Practice · Creases inside a region are not part of its perimeter · Sensim Math

Variant 1 easy answer: 46 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 10, 6 and 8 cm and height 9 cm is unfolded. The rectangle on the 10 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 10 cm, 6 cm and 8 cm.
The prism is 9 cm tall.
The hatched region is the 10 cm rectangle with a triangle on each of its 10 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 10, 6 or 8 for a base edge, 9 for an upright.

10,\ 6,\ 8,\ 9

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 10 cm wide, and each triangle is folded onto one of those 10 cm edges. Those two lines are creases inside the region, not part of its boundary.

10,\ 10 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

6 + 8 + 9 + 6 + 8 + 9

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(6 + 8) \times 2 + 9 \times 2 = 28 + 18 = 46

The hatched region is 46 cm around.

Answer: 46 cm

4 · Reviewdoes it hold up?

If the two 10 cm creases had been counted as well, the total would be 66 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 46, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 2 easy answer: 46 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 8, 5 and 6 cm and height 12 cm is unfolded. The rectangle on the 8 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 8 cm, 5 cm and 6 cm.
The prism is 12 cm tall.
The hatched region is the 8 cm rectangle with a triangle on each of its 8 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 8, 5 or 6 for a base edge, 12 for an upright.

8,\ 5,\ 6,\ 12

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 8 cm wide, and each triangle is folded onto one of those 8 cm edges. Those two lines are creases inside the region, not part of its boundary.

8,\ 8 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

5 + 6 + 12 + 5 + 6 + 12

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(5 + 6) \times 2 + 12 \times 2 = 22 + 24 = 46

The hatched region is 46 cm around.

Answer: 46 cm

4 · Reviewdoes it hold up?

If the two 8 cm creases had been counted as well, the total would be 62 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 46, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 3 easy answer: 56 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 12, 8 and 11 cm and height 9 cm is unfolded. The rectangle on the 12 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 12 cm, 8 cm and 11 cm.
The prism is 9 cm tall.
The hatched region is the 12 cm rectangle with a triangle on each of its 12 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 12, 8 or 11 for a base edge, 9 for an upright.

12,\ 8,\ 11,\ 9

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 12 cm wide, and each triangle is folded onto one of those 12 cm edges. Those two lines are creases inside the region, not part of its boundary.

12,\ 12 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

8 + 11 + 9 + 8 + 11 + 9

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(8 + 11) \times 2 + 9 \times 2 = 38 + 18 = 56

The hatched region is 56 cm around.

Answer: 56 cm

4 · Reviewdoes it hold up?

If the two 12 cm creases had been counted as well, the total would be 80 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 56, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 4 easy answer: 52 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 12, 7 and 9 cm and height 10 cm is unfolded. The rectangle on the 12 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 12 cm, 7 cm and 9 cm.
The prism is 10 cm tall.
The hatched region is the 12 cm rectangle with a triangle on each of its 12 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 12, 7 or 9 for a base edge, 10 for an upright.

12,\ 7,\ 9,\ 10

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 12 cm wide, and each triangle is folded onto one of those 12 cm edges. Those two lines are creases inside the region, not part of its boundary.

12,\ 12 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

7 + 9 + 10 + 7 + 9 + 10

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(7 + 9) \times 2 + 10 \times 2 = 32 + 20 = 52

The hatched region is 52 cm around.

Answer: 52 cm

4 · Reviewdoes it hold up?

If the two 12 cm creases had been counted as well, the total would be 76 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 52, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 5 medium answer: 58 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 13, 8 and 10 cm and height 11 cm is unfolded. The rectangle on the 13 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 13 cm, 8 cm and 10 cm.
The prism is 11 cm tall.
The hatched region is the 13 cm rectangle with a triangle on each of its 13 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 13, 8 or 10 for a base edge, 11 for an upright.

13,\ 8,\ 10,\ 11

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 13 cm wide, and each triangle is folded onto one of those 13 cm edges. Those two lines are creases inside the region, not part of its boundary.

13,\ 13 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

8 + 10 + 11 + 8 + 10 + 11

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(8 + 10) \times 2 + 11 \times 2 = 36 + 22 = 58

The hatched region is 58 cm around.

Answer: 58 cm

4 · Reviewdoes it hold up?

If the two 13 cm creases had been counted as well, the total would be 84 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 58, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 6 medium answer: 56 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 11, 6 and 9 cm and height 13 cm is unfolded. The rectangle on the 11 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 11 cm, 6 cm and 9 cm.
The prism is 13 cm tall.
The hatched region is the 11 cm rectangle with a triangle on each of its 11 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 11, 6 or 9 for a base edge, 13 for an upright.

11,\ 6,\ 9,\ 13

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 11 cm wide, and each triangle is folded onto one of those 11 cm edges. Those two lines are creases inside the region, not part of its boundary.

11,\ 11 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

6 + 9 + 13 + 6 + 9 + 13

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(6 + 9) \times 2 + 13 \times 2 = 30 + 26 = 56

The hatched region is 56 cm around.

Answer: 56 cm

4 · Reviewdoes it hold up?

If the two 11 cm creases had been counted as well, the total would be 78 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 56, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 7 medium answer: 54 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 14, 9 and 11 cm and height 7 cm is unfolded. The rectangle on the 14 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 14 cm, 9 cm and 11 cm.
The prism is 7 cm tall.
The hatched region is the 14 cm rectangle with a triangle on each of its 14 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 14, 9 or 11 for a base edge, 7 for an upright.

14,\ 9,\ 11,\ 7

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 14 cm wide, and each triangle is folded onto one of those 14 cm edges. Those two lines are creases inside the region, not part of its boundary.

14,\ 14 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

9 + 11 + 7 + 9 + 11 + 7

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(9 + 11) \times 2 + 7 \times 2 = 40 + 14 = 54

The hatched region is 54 cm around.

Answer: 54 cm

4 · Reviewdoes it hold up?

If the two 14 cm creases had been counted as well, the total would be 82 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 54, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 8 medium answer: 52 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 9, 5 and 7 cm and height 14 cm is unfolded. The rectangle on the 9 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 9 cm, 5 cm and 7 cm.
The prism is 14 cm tall.
The hatched region is the 9 cm rectangle with a triangle on each of its 9 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 9, 5 or 7 for a base edge, 14 for an upright.

9,\ 5,\ 7,\ 14

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 9 cm wide, and each triangle is folded onto one of those 9 cm edges. Those two lines are creases inside the region, not part of its boundary.

9,\ 9 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

5 + 7 + 14 + 5 + 7 + 14

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(5 + 7) \times 2 + 14 \times 2 = 24 + 28 = 52

The hatched region is 52 cm around.

Answer: 52 cm

4 · Reviewdoes it hold up?

If the two 9 cm creases had been counted as well, the total would be 70 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 52, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 9 hard answer: 48 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 7, 4 and 5 cm and height 15 cm is unfolded. The rectangle on the 7 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 7 cm, 4 cm and 5 cm.
The prism is 15 cm tall.
The hatched region is the 7 cm rectangle with a triangle on each of its 7 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 7, 4 or 5 for a base edge, 15 for an upright.

7,\ 4,\ 5,\ 15

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 7 cm wide, and each triangle is folded onto one of those 7 cm edges. Those two lines are creases inside the region, not part of its boundary.

7,\ 7 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

4 + 5 + 15 + 4 + 5 + 15

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(4 + 5) \times 2 + 15 \times 2 = 18 + 30 = 48

The hatched region is 48 cm around.

Answer: 48 cm

4 · Reviewdoes it hold up?

If the two 7 cm creases had been counted as well, the total would be 62 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 48, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 10 hard answer: 62 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 15, 10 and 13 cm and height 8 cm is unfolded. The rectangle on the 15 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 15 cm, 10 cm and 13 cm.
The prism is 8 cm tall.
The hatched region is the 15 cm rectangle with a triangle on each of its 15 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 15, 10 or 13 for a base edge, 8 for an upright.

15,\ 10,\ 13,\ 8

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 15 cm wide, and each triangle is folded onto one of those 15 cm edges. Those two lines are creases inside the region, not part of its boundary.

15,\ 15 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

10 + 13 + 8 + 10 + 13 + 8

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(10 + 13) \times 2 + 8 \times 2 = 46 + 16 = 62

The hatched region is 62 cm around.

Answer: 62 cm

4 · Reviewdoes it hold up?

If the two 15 cm creases had been counted as well, the total would be 92 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 62, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 11 hard answer: 58 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 16, 11 and 12 cm and height 6 cm is unfolded. The rectangle on the 16 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 16 cm, 11 cm and 12 cm.
The prism is 6 cm tall.
The hatched region is the 16 cm rectangle with a triangle on each of its 16 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 16, 11 or 12 for a base edge, 6 for an upright.

16,\ 11,\ 12,\ 6

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 16 cm wide, and each triangle is folded onto one of those 16 cm edges. Those two lines are creases inside the region, not part of its boundary.

16,\ 16 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

11 + 12 + 6 + 11 + 12 + 6

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(11 + 12) \times 2 + 6 \times 2 = 46 + 12 = 58

The hatched region is 58 cm around.

Answer: 58 cm

4 · Reviewdoes it hold up?

If the two 16 cm creases had been counted as well, the total would be 90 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 58, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Variant 12 hard answer: 62 cm

In the net of the triangular prism, what is the perimeter of the shaded part, in $\text{cm}$ ?

Show solution

1 · Understandwhat's really being asked

A triangular prism with base sides 18, 12 and 14 cm and height 5 cm is unfolded. The rectangle on the 18 cm side and the two triangles attached to it are hatched, and we want the distance round that hatched shape.

Givens

The base sides are 18 cm, 12 cm and 14 cm.
The prism is 5 cm tall.
The hatched region is the 18 cm rectangle with a triangle on each of its 18 cm sides.

Unknowns

The perimeter of the hatched region.

Constraints

Nothing in the net is measured directly; every segment is an edge of the solid.

2 · Planchoose the strategy

#17 Visualize Spatial Relationships · also uses: #1 Draw a Diagram#16 Count the Complement

Every line on a net is an edge of the solid, so each length is already one of the four given numbers. The only real question is which lines are on the outside of the hatched region -- walk its boundary and see.

3 · Execute4 carry out the plan

1Match the net's segments to the solid's edges

#17 Visualize Spatial Relationships 6.G.A.4

Every segment in a net is an edge of the solid folded flat, so each one is already known: 18, 12 or 14 for a base edge, 5 for an upright.

18,\ 12,\ 14,\ 5

Nothing needs measuring.

2Find which segments are inside the region

#16 Count the Complement 6.G.A.1

The rectangle in the hatched part is 18 cm wide, and each triangle is folded onto one of those 18 cm edges. Those two lines are creases inside the region, not part of its boundary.

18,\ 18 \rightarrow \text{inside}

A crease inside a shape is not on its edge.

3Walk the outside boundary

#1 Draw a Diagram 6.G.A.1

Going round: the two free sides of one triangle, an upright side of the rectangle, the two free sides of the other triangle, and the other upright side.

12 + 14 + 5 + 12 + 14 + 5

Six segments, in matching pairs.

4Add them

#1 Draw a Diagram 6.G.A.1

Group the matching pairs to keep the addition easy.

(12 + 14) \times 2 + 5 \times 2 = 52 + 10 = 62

The hatched region is 62 cm around.

Answer: 62 cm

4 · Reviewdoes it hold up?

If the two 18 cm creases had been counted as well, the total would be 98 -- so leaving them out is what the word 'perimeter' asks for.

Another way: Adding the three shapes' perimeters and then subtracting each shared edge twice gives the same 62, with more arithmetic.

Standardsmin grade 6

6.G.A.4 Surface area using nets of 3D figures — Reading each segment of the net as an edge of the prism.
6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Taking the perimeter of a composite region without its inner creases.

💡Takeaway. On a net you never have to measure: every line is already an edge you were given. Just decide which ones are on the outside.

Creases inside a region are not part of its perimeter · 12 practice problems

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