Practice · Perimeter is the sum of all side lengths · Sensim Math

Variant 1 easy

The perimeter of a rectangle is $9$ cm. If its width is $2\frac{1}{6}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $2\frac{1}{6}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $9$ cm and a width of $2\frac{1}{6}$ cm. We need its height.

Givens

Perimeter: $9$ cm.
Width: $2\frac{1}{6}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

2\frac{1}{6}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 2\frac{1}{6} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

2\frac{1}{6} + 2\frac{1}{6} = 4\frac{1}{3}

4\frac{1}{3}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

9 - 4\frac{1}{3} = 4\frac{2}{3}

4\frac{2}{3}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

4\frac{2}{3} \div 2 = 2\frac{1}{3}

The height is

2\frac{1}{3}

cm.

Answer:

2\frac{1}{3}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $2\frac{1}{6}$ + 2 x $2\frac{1}{3}$ = $9$ cm, which is what was given.

Another way: Halve the perimeter first: $9$ / 2 = $4\frac{1}{2}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 2 easy

The perimeter of a rectangle is $7\frac{5}{6}$ cm. If its width is $1\frac{2}{3}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $1\frac{2}{3}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $7\frac{5}{6}$ cm and a width of $1\frac{2}{3}$ cm. We need its height.

Givens

Perimeter: $7\frac{5}{6}$ cm.
Width: $1\frac{2}{3}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

1\frac{2}{3}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 1\frac{2}{3} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

1\frac{2}{3} + 1\frac{2}{3} = 3\frac{1}{3}

3\frac{1}{3}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

7\frac{5}{6} - 3\frac{1}{3} = 4\frac{1}{2}

4\frac{1}{2}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

4\frac{1}{2} \div 2 = 2\frac{1}{4}

The height is

2\frac{1}{4}

cm.

Answer:

2\frac{1}{4}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $1\frac{2}{3}$ + 2 x $2\frac{1}{4}$ = $7\frac{5}{6}$ cm, which is what was given.

Another way: Halve the perimeter first: $7\frac{5}{6}$ / 2 = $3\frac{11}{12}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 3 easy

The perimeter of a rectangle is $8\frac{1}{6}$ cm. If its width is $2\frac{1}{4}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $2\frac{1}{4}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $8\frac{1}{6}$ cm and a width of $2\frac{1}{4}$ cm. We need its height.

Givens

Perimeter: $8\frac{1}{6}$ cm.
Width: $2\frac{1}{4}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

2\frac{1}{4}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 2\frac{1}{4} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

2\frac{1}{4} + 2\frac{1}{4} = 4\frac{1}{2}

4\frac{1}{2}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

8\frac{1}{6} - 4\frac{1}{2} = 3\frac{2}{3}

3\frac{2}{3}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

3\frac{2}{3} \div 2 = 1\frac{5}{6}

The height is

1\frac{5}{6}

cm.

Answer:

1\frac{5}{6}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $2\frac{1}{4}$ + 2 x $1\frac{5}{6}$ = $8\frac{1}{6}$ cm, which is what was given.

Another way: Halve the perimeter first: $8\frac{1}{6}$ / 2 = $4\frac{1}{12}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 4 easy

The perimeter of a rectangle is $9\frac{1}{2}$ cm. If its width is $3\frac{1}{2}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $3\frac{1}{2}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $9\frac{1}{2}$ cm and a width of $3\frac{1}{2}$ cm. We need its height.

Givens

Perimeter: $9\frac{1}{2}$ cm.
Width: $3\frac{1}{2}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

3\frac{1}{2}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 3\frac{1}{2} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

3\frac{1}{2} + 3\frac{1}{2} = 7

7

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

9\frac{1}{2} - 7 = 2\frac{1}{2}

2\frac{1}{2}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

2\frac{1}{2} \div 2 = 1\frac{1}{4}

The height is

1\frac{1}{4}

cm.

Answer:

1\frac{1}{4}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $3\frac{1}{2}$ + 2 x $1\frac{1}{4}$ = $9\frac{1}{2}$ cm, which is what was given.

Another way: Halve the perimeter first: $9\frac{1}{2}$ / 2 = $4\frac{3}{4}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 5 medium

The perimeter of a rectangle is $9\frac{5}{6}$ cm. If its width is $3\frac{3}{4}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $3\frac{3}{4}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $9\frac{5}{6}$ cm and a width of $3\frac{3}{4}$ cm. We need its height.

Givens

Perimeter: $9\frac{5}{6}$ cm.
Width: $3\frac{3}{4}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

3\frac{3}{4}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 3\frac{3}{4} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

3\frac{3}{4} + 3\frac{3}{4} = 7\frac{1}{2}

7\frac{1}{2}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

9\frac{5}{6} - 7\frac{1}{2} = 2\frac{1}{3}

2\frac{1}{3}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

2\frac{1}{3} \div 2 = 1\frac{1}{6}

The height is

1\frac{1}{6}

cm.

Answer:

1\frac{1}{6}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $3\frac{3}{4}$ + 2 x $1\frac{1}{6}$ = $9\frac{5}{6}$ cm, which is what was given.

Another way: Halve the perimeter first: $9\frac{5}{6}$ / 2 = $4\frac{11}{12}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 6 medium

The perimeter of a rectangle is $10\frac{5}{6}$ cm. If its width is $3\frac{1}{6}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $3\frac{1}{6}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $10\frac{5}{6}$ cm and a width of $3\frac{1}{6}$ cm. We need its height.

Givens

Perimeter: $10\frac{5}{6}$ cm.
Width: $3\frac{1}{6}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

3\frac{1}{6}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 3\frac{1}{6} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

3\frac{1}{6} + 3\frac{1}{6} = 6\frac{1}{3}

6\frac{1}{3}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

10\frac{5}{6} - 6\frac{1}{3} = 4\frac{1}{2}

4\frac{1}{2}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

4\frac{1}{2} \div 2 = 2\frac{1}{4}

The height is

2\frac{1}{4}

cm.

Answer:

2\frac{1}{4}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $3\frac{1}{6}$ + 2 x $2\frac{1}{4}$ = $10\frac{5}{6}$ cm, which is what was given.

Another way: Halve the perimeter first: $10\frac{5}{6}$ / 2 = $5\frac{5}{12}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 7 medium

The perimeter of a rectangle is $8\frac{7}{10}$ cm. If its width is $2\frac{3}{4}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $2\frac{3}{4}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $8\frac{7}{10}$ cm and a width of $2\frac{3}{4}$ cm. We need its height.

Givens

Perimeter: $8\frac{7}{10}$ cm.
Width: $2\frac{3}{4}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

2\frac{3}{4}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 2\frac{3}{4} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

2\frac{3}{4} + 2\frac{3}{4} = 5\frac{1}{2}

5\frac{1}{2}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

8\frac{7}{10} - 5\frac{1}{2} = 3\frac{1}{5}

3\frac{1}{5}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

3\frac{1}{5} \div 2 = 1\frac{3}{5}

The height is

1\frac{3}{5}

cm.

Answer:

1\frac{3}{5}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $2\frac{3}{4}$ + 2 x $1\frac{3}{5}$ = $8\frac{7}{10}$ cm, which is what was given.

Another way: Halve the perimeter first: $8\frac{7}{10}$ / 2 = $4\frac{7}{20}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 8 medium

The perimeter of a rectangle is $10\frac{2}{3}$ cm. If its width is $2\frac{5}{6}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $2\frac{5}{6}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $10\frac{2}{3}$ cm and a width of $2\frac{5}{6}$ cm. We need its height.

Givens

Perimeter: $10\frac{2}{3}$ cm.
Width: $2\frac{5}{6}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

2\frac{5}{6}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 2\frac{5}{6} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

2\frac{5}{6} + 2\frac{5}{6} = 5\frac{2}{3}

5\frac{2}{3}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

10\frac{2}{3} - 5\frac{2}{3} = 5

5

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

5 \div 2 = 2\frac{1}{2}

The height is

2\frac{1}{2}

cm.

Answer:

2\frac{1}{2}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $2\frac{5}{6}$ + 2 x $2\frac{1}{2}$ = $10\frac{2}{3}$ cm, which is what was given.

Another way: Halve the perimeter first: $10\frac{2}{3}$ / 2 = $5\frac{1}{3}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 9 hard

The perimeter of a rectangle is $10\frac{2}{15}$ cm. If its width is $2\frac{2}{5}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $2\frac{2}{5}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $10\frac{2}{15}$ cm and a width of $2\frac{2}{5}$ cm. We need its height.

Givens

Perimeter: $10\frac{2}{15}$ cm.
Width: $2\frac{2}{5}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

2\frac{2}{5}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 2\frac{2}{5} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

2\frac{2}{5} + 2\frac{2}{5} = 4\frac{4}{5}

4\frac{4}{5}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

10\frac{2}{15} - 4\frac{4}{5} = 5\frac{1}{3}

5\frac{1}{3}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

5\frac{1}{3} \div 2 = 2\frac{2}{3}

The height is

2\frac{2}{3}

cm.

Answer:

2\frac{2}{3}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $2\frac{2}{5}$ + 2 x $2\frac{2}{3}$ = $10\frac{2}{15}$ cm, which is what was given.

Another way: Halve the perimeter first: $10\frac{2}{15}$ / 2 = $5\frac{1}{15}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 10 hard

The perimeter of a rectangle is $9\frac{11}{15}$ cm. If its width is $2\frac{2}{3}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $2\frac{2}{3}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $9\frac{11}{15}$ cm and a width of $2\frac{2}{3}$ cm. We need its height.

Givens

Perimeter: $9\frac{11}{15}$ cm.
Width: $2\frac{2}{3}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

2\frac{2}{3}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 2\frac{2}{3} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

2\frac{2}{3} + 2\frac{2}{3} = 5\frac{1}{3}

5\frac{1}{3}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

9\frac{11}{15} - 5\frac{1}{3} = 4\frac{2}{5}

4\frac{2}{5}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

4\frac{2}{5} \div 2 = 2\frac{1}{5}

The height is

2\frac{1}{5}

cm.

Answer:

2\frac{1}{5}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $2\frac{2}{3}$ + 2 x $2\frac{1}{5}$ = $9\frac{11}{15}$ cm, which is what was given.

Another way: Halve the perimeter first: $9\frac{11}{15}$ / 2 = $4\frac{13}{15}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 11 hard

The perimeter of a rectangle is $11\frac{11}{15}$ cm. If its width is $1\frac{1}{5}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $1\frac{1}{5}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $11\frac{11}{15}$ cm and a width of $1\frac{1}{5}$ cm. We need its height.

Givens

Perimeter: $11\frac{11}{15}$ cm.
Width: $1\frac{1}{5}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

1\frac{1}{5}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 1\frac{1}{5} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

1\frac{1}{5} + 1\frac{1}{5} = 2\frac{2}{5}

2\frac{2}{5}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

11\frac{11}{15} - 2\frac{2}{5} = 9\frac{1}{3}

9\frac{1}{3}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

9\frac{1}{3} \div 2 = 4\frac{2}{3}

The height is

4\frac{2}{3}

cm.

Answer:

4\frac{2}{3}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $1\frac{1}{5}$ + 2 x $4\frac{2}{3}$ = $11\frac{11}{15}$ cm, which is what was given.

Another way: Halve the perimeter first: $11\frac{11}{15}$ / 2 = $5\frac{13}{15}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Variant 12 hard

The perimeter of a rectangle is $7\frac{2}{15}$ cm. If its width is $1\frac{2}{5}$ cm, how many centimeters long is its height?

There is a rectangle whose top horizontal side is labeled $1\frac{2}{5}$ cm. The length of the vertical side is the value to find.

Your answer

Show solution

1 · Understandwhat's really being asked

A rectangle has a perimeter of $7\frac{2}{15}$ cm and a width of $1\frac{2}{5}$ cm. We need its height.

Givens

Perimeter: $7\frac{2}{15}$ cm.
Width: $1\frac{2}{5}$ cm.
A rectangle has two widths and two heights.

Unknowns

The height of the rectangle.

Constraints

Opposite sides of a rectangle are equal, so each length appears twice in the perimeter.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

Draw the rectangle and mark both widths. The perimeter is all four sides, so taking the two known ones away leaves the two heights -- and halving that gives one.

3 · Execute4 carry out the plan

1Draw and label the rectangle

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

Both horizontal sides are

1\frac{2}{5}

cm, and both vertical sides are the height we want.

\text{perimeter} = 2 \times 1\frac{2}{5} + 2 \times \text{height}

Four sides, but only two different lengths.

2Add the two widths

#11 Work Backwards 5.NF.A.1

The two horizontal sides together.

1\frac{2}{5} + 1\frac{2}{5} = 2\frac{4}{5}

2\frac{4}{5}

cm of the perimeter is accounted for.

3Subtract the widths from the perimeter to leave the two heights

#1 Draw a Diagram 5.NF.A.2

What remains of the perimeter is the two vertical sides.

7\frac{2}{15} - 2\frac{4}{5} = 4\frac{1}{3}

4\frac{1}{3}

cm split between two equal sides.

4Split the two heights into one height

#11 Work Backwards 5.NF.A.1

The two vertical sides are equal, so halve the remainder.

4\frac{1}{3} \div 2 = 2\frac{1}{6}

The height is

2\frac{1}{6}

cm.

Answer:

2\frac{1}{6}

cm

4 · Reviewdoes it hold up?

Rebuild the perimeter: 2 x $1\frac{2}{5}$ + 2 x $2\frac{1}{6}$ = $7\frac{2}{15}$ cm, which is what was given.

Another way: Halve the perimeter first: $7\frac{2}{15}$ / 2 = $3\frac{17}{30}$ cm is one width plus one height, and taking the width off leaves the height in one step.

Standardsmin grade 5

3.MD.D.8 Solve real world and mathematical problems involving perimeters of polygons, including finding an unknown side length — Reading the perimeter as the sum of all four sides.
5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) — Adding the two widths and halving the leftover.
5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole — Subtracting the known sides from the perimeter.

💡Takeaway. Every side of a rectangle appears twice in the perimeter, so take the pair you know away and halve what is left.

Perimeter is the sum of all side lengths · 12 practice problems

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