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Advanced Geometry : Trapezoids

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Example Questions

Example Question #31 : How To Find The Area Of A Trapezoid

Find the area of the figure below.

Possible Answers:

150.36

148.78

161.57

132.67

Correct answer:

161.57

Explanation:

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{7^2+12^2}=\sqrt{193}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(16+\sqrt{193})(8)=64+4\sqrt{193}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(7)(12)=42$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$

$\text{Area of Figure}=64+4\sqrt{193}+42=161.57$

Make sure to round to places after the decimal.

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Example Question #31 : How To Find The Area Of A Trapezoid

Find the area of the figure below.

Possible Answers:

50.67

55.30

62.12

59.38

Correct answer:

62.12

Explanation:

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{5^2+11^2}=\sqrt{146}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(11+\sqrt{146})(3)=\frac{33+3\sqrt{146}}{2}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(5)(11)=\frac{55}{2}$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$

$\text{Area of Figure}=\frac{33+3\sqrt{146}}{2}+\frac{55}{2}=62.12$

Make sure to round to places after the decimal.

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Example Question #33 : How To Find The Area Of A Trapezoid

Find the area of the figure below.

Possible Answers:

803.92

898.53

877.23

906.39

Correct answer:

898.53

Explanation:

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{22^2+26^2}=\sqrt{1160}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(34+\sqrt{1160})(18)=306+9\sqrt{1160}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(22)(26)=286$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$

$\text{Area of Figure}=306+9\sqrt{1160}+286=898.53$

Make sure to round to places after the decimal.

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Example Question #371 : Advanced Geometry

Find the area of the figure below.

Possible Answers:

150.81

140.58

131.95

136.01

Correct answer:

150.81

Explanation:

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{9^2+10^2}=\sqrt{181}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(13+\sqrt{181})(8)=52+4\sqrt{181}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(9)(10)=45$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$

$\text{Area of Figure}=52+4\sqrt{181}+45=150.81$

Make sure to round to places after the decimal.

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Example Question #31 : Trapezoids

Find the area of the figure.

Possible Answers:

91.85

94.37

103.37

111.73

Correct answer:

111.73

Explanation:

From the figure, you should notice that it is made up of a right triangle and a trapezoid. The lower base of the trapezoid is also the hypotenuse of the right triangle.

First, find the length of the hypotenuse of the right triangle using the Pythagorean Theorem.

$\text{Hypotenuse}=\sqrt{\text{base}^2+\text{height}^2}$

$\text{Hypotenuse}=\sqrt{7^2+12^2}=\sqrt{193}$

Next, use this value to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{1}{2}(\text{base 1}+\text{base 2})(\text{height})$

Plug in the given and found values to find the area.

$\text{Area of Trapezoid}=\frac{1}{2}(14+\sqrt{193})(5)=35+\frac{5\sqrt{193}}{2}$

Next, find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base})(\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(7)(12)=42$

To find the area of the figure, add the two areas together.

$\text{Area of Figure}=\text{Area of Trapezoid}+\text{Area of Triangle}$

$\text{Area of Figure}=35+\frac{5\sqrt{117}}{2}+42=111.73$

Make sure to round to places after the decimal.

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Example Question #32 : How To Find The Area Of A Trapezoid

Trapezoid

Figure NOT drawn to scale.

Examine the above trapezoid.

MN = 20 , TM = MP = 5 , RM = MA

True, false, or inconclusive: the area of Trapezoid TRAP is 200.

Possible Answers:

False

True

Correct answer:

True

Explanation:

The area of a trapezoid is equal to one half the product of half the height of the trapezoid and the sum of the lengths of the bases. This is

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

$A = \frac{1}{2} \cdot TP \cdot (TR + PA)$

or, equivalently,

$A = TP \cdot \frac{TR + PA}{2}$

The height of the trapezoid is

TP = TM + MP = 5 + 5 = 10 .

The lengths of bases $\overline{TR}$ and $\overline{PA}$ are not given, so it might appear that determining the area of the trapezoid is impossible.

However, it is given that TM = MP and RM = MA - that is, the segment $\overline{MN}$ bisects both legs of the trapezoid. This makes $\overline{MN}$ the midsegment of the trapezoid, the length of which is the arithmetic mean of those of the bases:

$MN = \frac{TR+ PA}{2}$ .

Therefore, the formula for the area of the trapezoid can be rewritten as

$A = TP \cdot MN$ ,

the product of the height and the length of the midsegment.

TP = 10 and MN = 20 , so

$A =10 \cdot 20 = 200$ ,

making the statement true.

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Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

What is the length of the diagonals of trapezoid ABCD ? Assume the figure is an isoceles trapezoid.

Trapezoid

Possible Answers:

$\sqrt{98}m$

$\sqrt{96}m$

$\sqrt{99}m$

$\sqrt{97}m$

$\sqrt{95}m$

Correct answer:

$\sqrt{97}m$

Explanation:

To find the length of the diagonal, we need to use the Pythagorean Theorem. Therefore, we need to sketch the following triangle within trapezoid ABCD :

Trapezoid

We know that the base of the triangle has length $9\: m$ . By subtracting the top of the trapezoid from the bottom of the trapezoid, we get:

$12\: m - 6\: m = 6\: m$

Dividing by two, we have the length of each additional side on the bottom of the trapezoid:

$\frac{6\: m}{2} = 3\: m$

Adding these two values together, we get $9\: m$ .

The formula for the length of diagonal uses the Pythagoreon Theorem:

AC^2 = AE^2 + EC^2 , where is the point between and representing the base of the triangle.

Plugging in our values, we get:

$AC^2 = (9\: m)^2 + (4\: m)^2$

$AC^2 = 81\: m^2 + 16\: m^2$

$AC^2 = 97\: m^2$

$AC = \sqrt{97}\: m$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of both diagonals of this quadrilateral.

Trapezoid 1

Possible Answers:

$\sqrt{97}, \sqrt{41}$

$\sqrt{97}, \sqrt{65}$

$\sqrt{85}, \sqrt{20}$

$\sqrt{41}, \sqrt{41}$

Correct answer:

$\sqrt{97}, \sqrt{65}$

Explanation:

All of the lengths with one mark have length 5, and all of the side lengths with two marks have length 4. With this knowledge, we can add side lengths together to find that one diagonal is the hypotenuse to this right triangle:

Trapezoid solution 3

Using Pythagorean Theorem gives:

4^2 + 9^2 = C^2

16+81=C^2

97 = C^2 take the square root of each side

$\sqrt{97}= C$

Similarly, the other diagonal can be found with this right triangle:

Trapezoid solution 4

Once again using Pythagorean Theorem gives an answer of $\sqrt{65}$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of the diagonals of this isosceles trapezoid, with $\textup{Height}=8$ .

Trapezoid 2

Possible Answers:

$\sqrt{73}$

$\sqrt{204}$

$\sqrt{128}$

$\sqrt{185}$

Correct answer:

$\sqrt{185}$

Explanation:

To find the length of the diagonals, split the top side into 3 sections as shown below:

Trapezoid solution 1

The two congruent sections plus 8 adds to 14. 14 - 8 = 6 , so the two congruent sections add to 6. They must each be 3. This means that the top of the right triangle with the diagonal as a hypotenuse must be 11, since 14-3=11 .

Trapezoid solution 2

We can solve for the diagonal, now pictured, using Pythagorean Theorem:

11^2 + 8^2 = C^2

121 + 64 = C^2

185 = C^2 take the square root of both sides

$\sqrt{185}=C$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Find the length of the diagonal of the isosceles trapezoid given below.

Trap1

Possible Answers:

$4\sqrt34\ cm$

$16\ cm$

$24\ cm$

$20\sqrt2\ cm$

Correct answer:

$4\sqrt34\ cm$

Explanation:

In order to calculate the length of the diagonal, we first must assume that the height is perpendicular to both the top and bottom of the trapezoid.

Knowing this, we can draw in the diagonal as shown below and use the Pythagorean Theorem to solve for the diagonal.

Trap2

$Using\ the\ pythagorean\ theorem\ we\ get: 12^2+20^2=diagonal^2$

$This\ simplifies\ to\ become\ 144+400=diagonal^2$

544=diagonal^2

We now take the square root of both sides:

$\sqrt544=\sqrt diagonal^2$

$\sqrt16*\sqrt34=diagonal$

$4\sqrt34=diagonal$

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Advanced Geometry : Trapezoids

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #31 : How To Find The Area Of A Trapezoid

Example Question #31 : How To Find The Area Of A Trapezoid

Example Question #33 : How To Find The Area Of A Trapezoid

Example Question #371 : Advanced Geometry

Example Question #31 : Trapezoids

Example Question #32 : How To Find The Area Of A Trapezoid

Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

Example Question #1 : How To Find The Length Of The Diagonal Of A Trapezoid

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