Area of a Trapezoid – Formulas and Examples

The area of a trapezoid can be considered as the region covered by the trapezoid in a two-dimensional plane. A trapezoid is a 2D figure that falls within the category of quadrilaterals. In a trapezoid, one pair of sides is parallel and the other pair of sides is not parallel. Similar to other geometric figures, the trapezoid also has its own properties and formulas based on area and perimeter.

Here, we will learn about the formula for the area of a trapezoid. In addition, we will look at some exercises in which we will apply this formula to find the area.

GEOMETRY
formula of the area of a trapezoid

Relevant for

Learning about the area of a trapezoid with examples.

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GEOMETRY
formula of the area of a trapezoid

Relevant for

Learning about the area of a trapezoid with examples.

See examples

Formula for the area of a trapezoid

To find the area of a trapezoid, we calculate the sum of its bases, multiply that sum by the height of the trapezoid, and then divide the result by 2. The formula for the area of the trapezoid is:

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

where,

  • $latex b_{1}=$ the length of the base 1 of the trapezoid
  • $latex b_{2}=$ the length of the base 2 of the trapezoid
  • $latex h=$ the length of the height of the trapezoid
diagram of the dimensions of a trapezoid

Each base of the trapezoid should be perpendicular to the height. In the figure, we can see that both bases are sides of the trapezoid. However, since the lateral sides are not perpendicular to either base, a broken line is drawn to represent the height.

Derivation of the formula for the area of a trapezoid

Let’s use the following figure to derive the formula for the area of a trapezoid:

diagram of the dimensions of a trapezoid 1

The area of a trapezoid is equal to the sum of the areas of the two triangles and the area of the rectangle. Thus, we know that:

trapezoid area = triangle area 1 + rectangle area + triangle area 2

$latex A=\frac{ah}{2}+b_{1}h+\frac{ch}{2}$

$latex A=\frac{ah+2b_{1}h+ch}{2}$

We can simplify the expression and rearrange the terms to get:

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

If we use $latex b_{2}$ to represent the longest base of the trapezoid, we have:

$latex b_{2}=a+b_{1}+c$

Substituting this in the previous equation, we have:

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

Therefore, the area of the trapezoid with bases $latex b_{1}, ~b_{2}$ and height $latex h$ is $latex A=\frac{h}{2}(b_{1}+b_{2})$.


Area of a trapezoid – Examples with answers

The following examples can be used to practice applying the formula for the area of a trapezoid. Try to solve the examples yourself before looking at the result.

EXAMPLE 1

Find the area of a trapezoid that has bases of length 8 m and 12 m and a height of 10 m.

We have the following information:

  • Base 1, $latex b_{1}=8$ m
  • Base 2, $latex b_{2}=12$ m
  • Height, $latex h=10$ m

Thus, we use the area formula by replacing the given values:

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

$latex =\frac{(8+12)10}{2}$

$latex =\frac{(20)(10)}{2}$

$latex =\frac{200}{2}$

$latex A=100$

The area of the trapezoid is 100 m².

EXAMPLE 2

What is the area of a trapezoid that has bases of length 11 m and 15 m and height 12 m?

The question gives us the following information:

  • Base 1, $latex b_{1}=11$ m
  • Base 2, $latex b_{2}=15$ m
  • Height, $latex h=12$ m

Using the given information, we have:

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

$latex =\frac{(11+15)12}{2}$

$latex =\frac{(26)(12)}{2}$

$latex =\frac{312}{2}$

$latex A=156$

The area of the trapezoid is 156 m².

EXAMPLE 3

A trapezoid has an area of 200 cm², a base of length 9 cm, and the other base of length 11 cm. What is the height of the trapezoid?

We have the following information:

  • Area, $latex A=200$ cm²
  • Base 1, $latex b_{1}=9$ cm
  • Base 2, $latex b_{2}=11$ cm

To find the height, we use the area formula and solve for h:

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

$latex 200=\frac{(9+11)h}{2}$

$latex 400=(9+11)h$

$latex 400=20h$

$latex h=20$

The height of the trapezoid is 20 cm.

EXAMPLE 4

A trapezoid has an area of 240 m², a base of length 11 m, and the other base of length 13 m. What is the height of the trapezoid?

From the question, we can get the following values:

  • Area, $latex A=240$ m²
  • Base 1, $latex b_{1}=11$ m
  • Base 2, $latex b_{2}=13$ m

We use the formula with the given values and solve for h:

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

$latex 240=\frac{(11+13)h}{2}$

$latex 480=(11+13)h$

$latex 480=24h$

$latex h=20$

The height of the trapezoid is 20 m.

EXAMPLE 5

Find the area of the following trapezoid:

diagram of the dimensions of a trapezoid example 5

Since the non-parallel sides of the trapezoid are equal, the height can be calculated as follows:

We get the bases of the two triangles by subtracting 7 from 15 and dividing by 2.

⇒ $latex \frac{15-7}{2}=4$ cm

diagram of the dimensions of a trapezoid example 51

Now, we can use the Pythagorean Theorem to find the height. Therefore, we have:

$latex {{8}^2}={{h}^2}+{{4}^2}$

$latex 64={{h}^2}+16$

$latex {{h}^2}=48$

$latex h=6.93$ cm

The height of the trapezoid is 6.93 cm. Now, we calculate the area with these values:

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

$latex =\frac{(15+7)6.93}{2}$

$latex =\frac{(22)(6.93)}{2}$

$latex A=76.23$

The area of the trapezoid is 76.23 cm².


Area of a trapezoid – Practice problems

Put into practice what you have learned about the area of a trapezoid to solve the following problems. Select an answer and check it to make sure you selected the correct one.

Find the area of a trapezoid that has bases of length 6 m and 10 m and a height of 5 m.

Choose an answer






Find the area of a trapezoid that has bases of length 13 m and 15 m and a height of 8 m.

Choose an answer






What is the height of a trapezoid that has an area of 322 $latex {{m}^2}$ and bases of length 19 m and 27 m?

Choose an answer






What is the dimension of the other base of a trapezoid that has an area of 352 $latex {{m}^2}$, a base of 25 m, and a height of 16 m?

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