How to find the area of a trapezoid - ISEE Upper Level Mathematics Achievement

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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

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

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

$b_{1}=a=12\ cm$

$b_{2}=b=14\ cm$

$h=10\ cm$

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{12+14}{2})\times 10=13\times 10=130\ cm^2$

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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

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

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

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{3t+t}{2})\times h=8t^2\Rightarrow (2t)h=8t^2$

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

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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

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

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

$b_{1}=a$

$b_{2}=b=2a$

Substitute these values into the area formula:

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{a+2a}{2})a=54\Rightarrow \frac{3a^2}{2}=54\Rightarrow a^2=36\Rightarrow a=6\ in$

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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 $b_{1} = SR = 24$ and $b_{2} = QU$ , 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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