← Composite perimeter sums boundary edges · Perimeter by Tracing Every Side

Composite perimeter sums boundary edges · 12 practice problems

3.MD.D.83.OA.A.2

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 72 cm

The figure ABCDE on the right is made up of one equilateral triangle and $2$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Two congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $18\ \text{cm}$ , which is the combined width of the $2$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 2 equal squares in a row with an equilateral triangle on top. The row is 18 cm wide. We need the distance all the way around the outside.

Givens

2 congruent squares sit side by side.
Their combined bottom edge is 18 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 2 congruent squares share the 18 cm bottom equally, so divide 18 by 2 to get the side of one square.

18 \div 2 = 9 \text{ cm}

Every square is 9 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 18 cm. Because the triangle is equilateral, its two slanted sides are also 18 cm each.

\text{each triangle side} = 18 \text{ cm}

Both slanted sides are 18 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 18 cm each), the left side of the row (B-C, one square's side = 9 cm), the bottom (C-D = 18 cm), and the right side of the row (D-E = 9 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 18,\ 9,\ 18,\ 9,\ 18 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

18 + 9 + 18 + 9 + 18 = 72 \text{ cm}

The way around is 72 cm.

Answer: 72 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 18 (3 x 18 = 54) plus two of length 9 (2 x 9 = 18), and 54 + 18 = 72 cm.

Another way: Adding every edge of every piece would give far more than 72 cm, because it would count the 1 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 2 easy answer: 77 cm

The figure ABCDE on the right is made up of one equilateral triangle and $3$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Three congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $21\ \text{cm}$ , which is the combined width of the $3$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 3 equal squares in a row with an equilateral triangle on top. The row is 21 cm wide. We need the distance all the way around the outside.

Givens

3 congruent squares sit side by side.
Their combined bottom edge is 21 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 3 congruent squares share the 21 cm bottom equally, so divide 21 by 3 to get the side of one square.

21 \div 3 = 7 \text{ cm}

Every square is 7 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 21 cm. Because the triangle is equilateral, its two slanted sides are also 21 cm each.

\text{each triangle side} = 21 \text{ cm}

Both slanted sides are 21 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 21 cm each), the left side of the row (B-C, one square's side = 7 cm), the bottom (C-D = 21 cm), and the right side of the row (D-E = 7 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 21,\ 7,\ 21,\ 7,\ 21 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

21 + 7 + 21 + 7 + 21 = 77 \text{ cm}

The way around is 77 cm.

Answer: 77 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 21 (3 x 21 = 63) plus two of length 7 (2 x 7 = 14), and 63 + 14 = 77 cm.

Another way: Adding every edge of every piece would give far more than 77 cm, because it would count the 2 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 3 easy answer: 104 cm

The figure ABCDE on the right is made up of one equilateral triangle and $2$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Two congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $26\ \text{cm}$ , which is the combined width of the $2$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 2 equal squares in a row with an equilateral triangle on top. The row is 26 cm wide. We need the distance all the way around the outside.

Givens

2 congruent squares sit side by side.
Their combined bottom edge is 26 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 2 congruent squares share the 26 cm bottom equally, so divide 26 by 2 to get the side of one square.

26 \div 2 = 13 \text{ cm}

Every square is 13 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 26 cm. Because the triangle is equilateral, its two slanted sides are also 26 cm each.

\text{each triangle side} = 26 \text{ cm}

Both slanted sides are 26 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 26 cm each), the left side of the row (B-C, one square's side = 13 cm), the bottom (C-D = 26 cm), and the right side of the row (D-E = 13 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 26,\ 13,\ 26,\ 13,\ 26 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

26 + 13 + 26 + 13 + 26 = 104 \text{ cm}

The way around is 104 cm.

Answer: 104 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 26 (3 x 26 = 78) plus two of length 13 (2 x 13 = 26), and 78 + 26 = 104 cm.

Another way: Adding every edge of every piece would give far more than 104 cm, because it would count the 1 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 4 easy answer: 110 cm

The figure ABCDE on the right is made up of one equilateral triangle and $3$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Three congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $30\ \text{cm}$ , which is the combined width of the $3$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 3 equal squares in a row with an equilateral triangle on top. The row is 30 cm wide. We need the distance all the way around the outside.

Givens

3 congruent squares sit side by side.
Their combined bottom edge is 30 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 3 congruent squares share the 30 cm bottom equally, so divide 30 by 3 to get the side of one square.

30 \div 3 = 10 \text{ cm}

Every square is 10 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 30 cm. Because the triangle is equilateral, its two slanted sides are also 30 cm each.

\text{each triangle side} = 30 \text{ cm}

Both slanted sides are 30 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 30 cm each), the left side of the row (B-C, one square's side = 10 cm), the bottom (C-D = 30 cm), and the right side of the row (D-E = 10 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 30,\ 10,\ 30,\ 10,\ 30 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

30 + 10 + 30 + 10 + 30 = 110 \text{ cm}

The way around is 110 cm.

Answer: 110 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 30 (3 x 30 = 90) plus two of length 10 (2 x 10 = 20), and 90 + 20 = 110 cm.

Another way: Adding every edge of every piece would give far more than 110 cm, because it would count the 2 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 5 medium answer: 126 cm

The figure ABCDE on the right is made up of one equilateral triangle and $4$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Four congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $36\ \text{cm}$ , which is the combined width of the $4$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 4 equal squares in a row with an equilateral triangle on top. The row is 36 cm wide. We need the distance all the way around the outside.

Givens

4 congruent squares sit side by side.
Their combined bottom edge is 36 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 4 congruent squares share the 36 cm bottom equally, so divide 36 by 4 to get the side of one square.

36 \div 4 = 9 \text{ cm}

Every square is 9 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 36 cm. Because the triangle is equilateral, its two slanted sides are also 36 cm each.

\text{each triangle side} = 36 \text{ cm}

Both slanted sides are 36 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 36 cm each), the left side of the row (B-C, one square's side = 9 cm), the bottom (C-D = 36 cm), and the right side of the row (D-E = 9 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 36,\ 9,\ 36,\ 9,\ 36 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

36 + 9 + 36 + 9 + 36 = 126 \text{ cm}

The way around is 126 cm.

Answer: 126 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 36 (3 x 36 = 108) plus two of length 9 (2 x 9 = 18), and 108 + 18 = 126 cm.

Another way: Adding every edge of every piece would give far more than 126 cm, because it would count the 3 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 6 medium answer: 136 cm

The figure ABCDE on the right is made up of one equilateral triangle and $5$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Five congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $40\ \text{cm}$ , which is the combined width of the $5$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 5 equal squares in a row with an equilateral triangle on top. The row is 40 cm wide. We need the distance all the way around the outside.

Givens

5 congruent squares sit side by side.
Their combined bottom edge is 40 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 5 congruent squares share the 40 cm bottom equally, so divide 40 by 5 to get the side of one square.

40 \div 5 = 8 \text{ cm}

Every square is 8 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 40 cm. Because the triangle is equilateral, its two slanted sides are also 40 cm each.

\text{each triangle side} = 40 \text{ cm}

Both slanted sides are 40 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 40 cm each), the left side of the row (B-C, one square's side = 8 cm), the bottom (C-D = 40 cm), and the right side of the row (D-E = 8 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 40,\ 8,\ 40,\ 8,\ 40 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

40 + 8 + 40 + 8 + 40 = 136 \text{ cm}

The way around is 136 cm.

Answer: 136 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 40 (3 x 40 = 120) plus two of length 8 (2 x 8 = 16), and 120 + 16 = 136 cm.

Another way: Adding every edge of every piece would give far more than 136 cm, because it would count the 4 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 7 medium answer: 165 cm

The figure ABCDE on the right is made up of one equilateral triangle and $3$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Three congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $45\ \text{cm}$ , which is the combined width of the $3$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 3 equal squares in a row with an equilateral triangle on top. The row is 45 cm wide. We need the distance all the way around the outside.

Givens

3 congruent squares sit side by side.
Their combined bottom edge is 45 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 3 congruent squares share the 45 cm bottom equally, so divide 45 by 3 to get the side of one square.

45 \div 3 = 15 \text{ cm}

Every square is 15 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 45 cm. Because the triangle is equilateral, its two slanted sides are also 45 cm each.

\text{each triangle side} = 45 \text{ cm}

Both slanted sides are 45 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 45 cm each), the left side of the row (B-C, one square's side = 15 cm), the bottom (C-D = 45 cm), and the right side of the row (D-E = 15 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 45,\ 15,\ 45,\ 15,\ 45 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

45 + 15 + 45 + 15 + 45 = 165 \text{ cm}

The way around is 165 cm.

Answer: 165 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 45 (3 x 45 = 135) plus two of length 15 (2 x 15 = 30), and 135 + 30 = 165 cm.

Another way: Adding every edge of every piece would give far more than 165 cm, because it would count the 2 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 8 medium answer: 168 cm

The figure ABCDE on the right is made up of one equilateral triangle and $4$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Four congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $48\ \text{cm}$ , which is the combined width of the $4$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 4 equal squares in a row with an equilateral triangle on top. The row is 48 cm wide. We need the distance all the way around the outside.

Givens

4 congruent squares sit side by side.
Their combined bottom edge is 48 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 4 congruent squares share the 48 cm bottom equally, so divide 48 by 4 to get the side of one square.

48 \div 4 = 12 \text{ cm}

Every square is 12 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 48 cm. Because the triangle is equilateral, its two slanted sides are also 48 cm each.

\text{each triangle side} = 48 \text{ cm}

Both slanted sides are 48 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 48 cm each), the left side of the row (B-C, one square's side = 12 cm), the bottom (C-D = 48 cm), and the right side of the row (D-E = 12 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 48,\ 12,\ 48,\ 12,\ 48 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

48 + 12 + 48 + 12 + 48 = 168 \text{ cm}

The way around is 168 cm.

Answer: 168 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 48 (3 x 48 = 144) plus two of length 12 (2 x 12 = 24), and 144 + 24 = 168 cm.

Another way: Adding every edge of every piece would give far more than 168 cm, because it would count the 3 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 9 hard answer: 180 cm

The figure ABCDE on the right is made up of one equilateral triangle and $6$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Six congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $54\ \text{cm}$ , which is the combined width of the $6$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 equal squares in a row with an equilateral triangle on top. The row is 54 cm wide. We need the distance all the way around the outside.

Givens

6 congruent squares sit side by side.
Their combined bottom edge is 54 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 6 congruent squares share the 54 cm bottom equally, so divide 54 by 6 to get the side of one square.

54 \div 6 = 9 \text{ cm}

Every square is 9 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 54 cm. Because the triangle is equilateral, its two slanted sides are also 54 cm each.

\text{each triangle side} = 54 \text{ cm}

Both slanted sides are 54 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 54 cm each), the left side of the row (B-C, one square's side = 9 cm), the bottom (C-D = 54 cm), and the right side of the row (D-E = 9 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 54,\ 9,\ 54,\ 9,\ 54 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

54 + 9 + 54 + 9 + 54 = 180 \text{ cm}

The way around is 180 cm.

Answer: 180 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 54 (3 x 54 = 162) plus two of length 9 (2 x 9 = 18), and 162 + 18 = 180 cm.

Another way: Adding every edge of every piece would give far more than 180 cm, because it would count the 5 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 10 hard answer: 210 cm

The figure ABCDE on the right is made up of one equilateral triangle and $4$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Four congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $60\ \text{cm}$ , which is the combined width of the $4$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 4 equal squares in a row with an equilateral triangle on top. The row is 60 cm wide. We need the distance all the way around the outside.

Givens

4 congruent squares sit side by side.
Their combined bottom edge is 60 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 4 congruent squares share the 60 cm bottom equally, so divide 60 by 4 to get the side of one square.

60 \div 4 = 15 \text{ cm}

Every square is 15 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 60 cm. Because the triangle is equilateral, its two slanted sides are also 60 cm each.

\text{each triangle side} = 60 \text{ cm}

Both slanted sides are 60 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 60 cm each), the left side of the row (B-C, one square's side = 15 cm), the bottom (C-D = 60 cm), and the right side of the row (D-E = 15 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 60,\ 15,\ 60,\ 15,\ 60 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

60 + 15 + 60 + 15 + 60 = 210 \text{ cm}

The way around is 210 cm.

Answer: 210 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 60 (3 x 60 = 180) plus two of length 15 (2 x 15 = 30), and 180 + 30 = 210 cm.

Another way: Adding every edge of every piece would give far more than 210 cm, because it would count the 3 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 11 hard answer: 221 cm

The figure ABCDE on the right is made up of one equilateral triangle and $5$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Five congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $65\ \text{cm}$ , which is the combined width of the $5$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 5 equal squares in a row with an equilateral triangle on top. The row is 65 cm wide. We need the distance all the way around the outside.

Givens

5 congruent squares sit side by side.
Their combined bottom edge is 65 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 5 congruent squares share the 65 cm bottom equally, so divide 65 by 5 to get the side of one square.

65 \div 5 = 13 \text{ cm}

Every square is 13 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 65 cm. Because the triangle is equilateral, its two slanted sides are also 65 cm each.

\text{each triangle side} = 65 \text{ cm}

Both slanted sides are 65 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 65 cm each), the left side of the row (B-C, one square's side = 13 cm), the bottom (C-D = 65 cm), and the right side of the row (D-E = 13 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 65,\ 13,\ 65,\ 13,\ 65 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

65 + 13 + 65 + 13 + 65 = 221 \text{ cm}

The way around is 221 cm.

Answer: 221 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 65 (3 x 65 = 195) plus two of length 13 (2 x 13 = 26), and 195 + 26 = 221 cm.

Another way: Adding every edge of every piece would give far more than 221 cm, because it would count the 4 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.

Variant 12 hard answer: 240 cm

The figure ABCDE on the right is made up of one equilateral triangle and $6$ congruent squares of the same size. What is the perimeter of figure ABCDE, in $\text{cm}$ ?

Figure description: Six congruent squares sit side by side in a row, and one equilateral triangle rests on top of them. The bottom edge of the row (the side from the lower-left to the lower-right corner) measures $72\ \text{cm}$ , which is the combined width of the $6$ squares. The apex of the triangle is at the top, the triangle's base runs along the full top edge of the row of squares, and the two lower corners complete the outline.

Show solution

1 · Understandwhat's really being asked

A shape is built from 6 equal squares in a row with an equilateral triangle on top. The row is 72 cm wide. We need the distance all the way around the outside.

Givens

6 congruent squares sit side by side.
Their combined bottom edge is 72 cm.
The triangle on top is equilateral and its base is the whole top of the row.

Unknowns

The perimeter of the whole figure ABCDE.

Constraints

Only the outside edges count; the lines where pieces meet are inside the figure.

2 · Planchoose the strategy

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Find the one square's side first, because every short edge in the figure is that length. Then walk around the outline once and add only the edges you actually walk along.

3 · Execute4 carry out the plan

1Find one square's side length

#7 Identify Subproblems 3.OA.A.2

The 6 congruent squares share the 72 cm bottom equally, so divide 72 by 6 to get the side of one square.

72 \div 6 = 12 \text{ cm}

Every square is 12 cm on a side.

2Find each side of the equilateral triangle

#7 Identify Subproblems 3.MD.D.8

The triangle's base lies along the whole top edge of the row, so the base is 72 cm. Because the triangle is equilateral, its two slanted sides are also 72 cm each.

\text{each triangle side} = 72 \text{ cm}

Both slanted sides are 72 cm.

3Trace the outside boundary

#1 Draw a Diagram 3.MD.D.8

Walking around ABCDE the outside edges are: the two slanted triangle sides (A-B and E-A, 72 cm each), the left side of the row (B-C, one square's side = 12 cm), the bottom (C-D = 72 cm), and the right side of the row (D-E = 12 cm). The line where the triangle meets the row, and the lines between squares, are inside.

\text{outside edges} = 72,\ 12,\ 72,\ 12,\ 72 \text{ (cm)}

Five edges, none of them internal.

4Add the boundary edges

#7 Identify Subproblems 3.MD.D.8

Add the five outside edge lengths to get the perimeter.

72 + 12 + 72 + 12 + 72 = 240 \text{ cm}

The way around is 240 cm.

Answer: 240 cm

4 · Reviewdoes it hold up?

Grouped another way: three edges of length 72 (3 x 72 = 216) plus two of length 12 (2 x 12 = 24), and 216 + 24 = 240 cm.

Another way: Adding every edge of every piece would give far more than 240 cm, because it would count the 5 lines between squares and the triangle's base -- edges you never walk along.

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the outside edges of a composite figure.
3.OA.A.2 Interpret whole-number quotients of whole numbers — Sharing the bottom edge equally among the congruent squares.

💡Takeaway. Perimeter is the walk around the outside, so the lines where two pieces touch never count.