How to find the area of a trapezoid - ISEE Upper Level Quantitative Reasoning

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Question

Find the area of the above trapezoid if $a=12\ cm$ , $b=14\ cm$ , and $h=10\ cm$ .

Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

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Question

A trapezoid has the base lengths of and . The area of the trapezoid is . Give the height of the trapezoid in terms of .

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$\Rightarrow $h=\frac{8t^2$}{2t}$=4t$

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Question

In the following trapezoid and . The area of the trapezoid is 54 square inches. Give the height of the trapezoid. Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

Substitute these values into the area formula:

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Question

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

Answer

Substitute in the formula for the area of a trapezoid:

$$\frac{1}{2}$ (B + b)h = A$

$$\frac{1}{2}$ (16 + 12)h = 98$

$$\frac{1}{2}$ (28)h = 98$

$14h \div 14= 98 \div 14$

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Question

A trapezoid has height 20 inches. Its bases have sum 30 inches, and one base is 6 inches longer than the other. What is the area of this trapezoid?

Answer

You do not need to find the individual bases; their sum, , is . You can substitute into the formula for the area of a trapezoid:

$A = $\frac{1}{2}$ (B + b)h$

$A = $\frac{1}{2}$\cdot 30\cdot 20 = 300$ square inches.

Note that the fact that one base is inches longer is not important here.

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Question

What is the area of the shaded portion of the above square?

Answer

Quadrilateral - the shaded region - is a trapezoid with bases $\overline{SR}$ and $\overline{QU }$ , and altitude $\overline{SQ}$ . The area of the trapezoid can be calculated using the formula

$A = $\frac{1}{2}$ ( $b_{1}$+ $b_{2}$ ) h$ ,

where and , and .

The length of $\overline{UA}$ can be found by setting and and applying the Pythagorean Theorem:

$a = $$\sqrt{c^{2}$$ - $b^{2}$} = $$\sqrt{25^{2}$$ - $24^{2}$} =$\sqrt{625- 576}$ = $\sqrt{49}$ = 7$

Therefore,

.

Substituting:

$A = $\frac{1}{2}$ ( $b_{1}$+ $b_{2}$ ) h$

$A = $\frac{1}{2}$ ( 24 + 17) 24$

$A = $\frac{1}{2}$ \cdot 41 \cdot 24$

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Question

Find the area of the above trapezoid if $a=12\ cm$ , $b=14\ cm$ , and $h=10\ cm$ .

Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

Compare your answer with the correct one above

Question

A trapezoid has the base lengths of and . The area of the trapezoid is . Give the height of the trapezoid in terms of .

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$\Rightarrow $h=\frac{8t^2$}{2t}$=4t$

Compare your answer with the correct one above

Question

In the following trapezoid and . The area of the trapezoid is 54 square inches. Give the height of the trapezoid. Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

Substitute these values into the area formula:

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

Answer

Substitute in the formula for the area of a trapezoid:

$$\frac{1}{2}$ (B + b)h = A$

$$\frac{1}{2}$ (16 + 12)h = 98$

$$\frac{1}{2}$ (28)h = 98$

$14h \div 14= 98 \div 14$

Compare your answer with the correct one above

Question

A trapezoid has height 20 inches. Its bases have sum 30 inches, and one base is 6 inches longer than the other. What is the area of this trapezoid?

Answer

You do not need to find the individual bases; their sum, , is . You can substitute into the formula for the area of a trapezoid:

$A = $\frac{1}{2}$ (B + b)h$

$A = $\frac{1}{2}$\cdot 30\cdot 20 = 300$ square inches.

Note that the fact that one base is inches longer is not important here.

Compare your answer with the correct one above

Question

What is the area of the shaded portion of the above square?

Answer

Quadrilateral - the shaded region - is a trapezoid with bases $\overline{SR}$ and $\overline{QU }$ , and altitude $\overline{SQ}$ . The area of the trapezoid can be calculated using the formula

$A = $\frac{1}{2}$ ( $b_{1}$+ $b_{2}$ ) h$ ,

where and , and .

The length of $\overline{UA}$ can be found by setting and and applying the Pythagorean Theorem:

$a = $$\sqrt{c^{2}$$ - $b^{2}$} = $$\sqrt{25^{2}$$ - $24^{2}$} =$\sqrt{625- 576}$ = $\sqrt{49}$ = 7$

Therefore,

.

Substituting:

$A = $\frac{1}{2}$ ( $b_{1}$+ $b_{2}$ ) h$

$A = $\frac{1}{2}$ ( 24 + 17) 24$

$A = $\frac{1}{2}$ \cdot 41 \cdot 24$

Compare your answer with the correct one above

Question

Find the area of the above trapezoid if $a=12\ cm$ , $b=14\ cm$ , and $h=10\ cm$ .

Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

Compare your answer with the correct one above

Question

A trapezoid has the base lengths of and . The area of the trapezoid is . Give the height of the trapezoid in terms of .

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$\Rightarrow $h=\frac{8t^2$}{2t}$=4t$

Compare your answer with the correct one above

Question

In the following trapezoid and . The area of the trapezoid is 54 square inches. Give the height of the trapezoid. Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

Substitute these values into the area formula:

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

Answer

Substitute in the formula for the area of a trapezoid:

$$\frac{1}{2}$ (B + b)h = A$

$$\frac{1}{2}$ (16 + 12)h = 98$

$$\frac{1}{2}$ (28)h = 98$

$14h \div 14= 98 \div 14$

Compare your answer with the correct one above

Question

A trapezoid has height 20 inches. Its bases have sum 30 inches, and one base is 6 inches longer than the other. What is the area of this trapezoid?

Answer

You do not need to find the individual bases; their sum, , is . You can substitute into the formula for the area of a trapezoid:

$A = $\frac{1}{2}$ (B + b)h$

$A = $\frac{1}{2}$\cdot 30\cdot 20 = 300$ square inches.

Note that the fact that one base is inches longer is not important here.

Compare your answer with the correct one above

Question

What is the area of the shaded portion of the above square?

Answer

Quadrilateral - the shaded region - is a trapezoid with bases $\overline{SR}$ and $\overline{QU }$ , and altitude $\overline{SQ}$ . The area of the trapezoid can be calculated using the formula

$A = $\frac{1}{2}$ ( $b_{1}$+ $b_{2}$ ) h$ ,

where and , and .

The length of $\overline{UA}$ can be found by setting and and applying the Pythagorean Theorem:

$a = $$\sqrt{c^{2}$$ - $b^{2}$} = $$\sqrt{25^{2}$$ - $24^{2}$} =$\sqrt{625- 576}$ = $\sqrt{49}$ = 7$

Therefore,

.

Substituting:

$A = $\frac{1}{2}$ ( $b_{1}$+ $b_{2}$ ) h$

$A = $\frac{1}{2}$ ( 24 + 17) 24$

$A = $\frac{1}{2}$ \cdot 41 \cdot 24$

Compare your answer with the correct one above

Question

Find the area of the above trapezoid if $a=12\ cm$ , $b=14\ cm$ , and $h=10\ cm$ .

Figure not drawn to scale.

Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

Compare your answer with the correct one above

Question

A trapezoid has the base lengths of and . The area of the trapezoid is . Give the height of the trapezoid in terms of .

Answer

The area of a trapezoid is given by

,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$\Rightarrow $h=\frac{8t^2$}{2t}$=4t$

Compare your answer with the correct one above

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