## What is the Area of a Trapezoid Formula?

The area of a trapezoid formula is…

\[A=\frac{1}{2}h(b_1 + b_2)\] The \(b_1\) and \(b_2\) variables represent the lengths of the bases of the trapezoid. The bases of the trapezoid must be parallel to each other.

The \(h\) variable represents the height of the trapezoid. The height of the trapezoid must be perpendicular to the bases.

## How to Use the Area of a Trapezoid Formula

- Identify what measurements you were given.
- Identify which measurement you were asked to find.
- Substitute the given measurements into the Area of a Trapezoid Formula. Then simplify or solve the equation until you find the measurement you were asked to find.

## Examples

These examples are advanced applications of the Area of a Trapezoid Formula. If you want to see some basic examples of how to find the area of a trapezoid, check out this page.

Find the missing base of the trapezoid if the area of the trapezoid is \(16.5 \; in^2\).

*What measurements were given to you?*

The problem told me that the area of the trapezoid is \(16.5 \; in^2\).

\[\yellow A=16.5\]

The diagram tells me that one of the bases is \(4 \; in\) and the height of the trapezoid is \(3 \; in\).

\[\yellow b_1=4\]

\[\yellow h=3\]

*Which measurement are you looking for?*

The measurement I want to find is the length of the missing base.

\[\yellow b_2 = b\]

*Do you have all the variables required for the Area of a Trapezoid Formula?*

Yes, in Steps 1 and 2, I defined all four variables:

\[\yellow A=16.5\]

\[\yellow b_1=4\]

\[\yellow h=3\]

\[\yellow b_2 = b\]

When I substitute these values into the Area of a Trapezoid Formula, I get…

\[{\yellow 16.5}=\frac{1}{2}({\yellow 3})({\yellow 4}+{\yellow b})\]

I can simplify the right side of the equation by multiplying \(\frac{1}{2}\) by \(\yellow 3\).

\[{\yellow 16.5}=1.5({\yellow 4}+{\yellow b})\]

Then I can divide both sides of the equation by \(1.5\).

\[11={\yellow 4}+{\yellow b}\]

Lastly, I will subtract \(\yellow 4\) from both sides of the equation to isolate the \(\yellow b\).

\[8={\yellow b}\]

The missing base of the trapezoid is \(8 \; in\) long.

Find the missing height of the trapezoid if the area of the trapezoid is \(57.5 \; cm^2\).

*What measurements were given to you?*

The problem told me that the area of the trapezoid is \(57.5 \; cm^2\).

\[\blue A=57.5\]

The diagram tells me that the bases are \(15 \; cm\) and \(8 \; cm\) long.

\[\blue b_1=15\]

\[\blue b_2=8\]

*Which measurement are you looking for?*

The measurement I want to find is the height of the trapezoid.

\[\blue h = h\]

*Do you have all the variables required for the Area of a Trapezoid Formula?*

Yes, in Steps 1 and 2, I defined all four variables:

\[\blue A=57.5\]

\[\blue b_1=15\]

\[\blue b_2=8\]

\[\blue h=h\]

When I substitute these values into the Area of a Trapezoid Formula, I get…

\[{\blue 57.5}=\frac{1}{2}({\blue h})({\blue 15}+{\blue 8})\]

I can simplify the right side of the equation by adding \(\blue 15\) and \(\blue 8\).

\[{\blue 57.5}=\frac{1}{2}({\blue h})(23)\]

Then I can multiply \(\frac{1}{2}\) by \(23\).

\[{\blue 57.5}=11.5{\blue h}\]

Lastly, I will divide both sides of the equation by \(11.5\) to isolate the \(\blue h\).

[5={\blue h}\]

The height of the trapezoid is \(5 \; cm\) long.

Find the area of the trapezoid.

*What measurements were given to you?*

The diagram tells me that one of the bases of the trapezoid is \(3 \; ft\).

\[\purple b_1=3\]

The diagram also tells me that the other base is split into two parts measuring \(3 \; ft\) and \(7 \; ft\). I can add these together to find the length of the base.

\[\purple b_2=10\]

The diagram does not explicitly give me the height of the trapezoid, but the height, the slant height \(5 \; ft\) and the partial base \(3 \; ft\) form a right triangle so I can use the Pythagorean Theorem to find the height of the trapezoid.

\[\purple 3^2 + h^2= 5^2\]

\[\purple 9 + h^2= 25\]

\[\purple h^2= 16\]

\[\purple h= 4\]

*Which measurement are you looking for?*

The measurement I want to find is the area of the trapezoid.

\[\purple A = A\]

*Do you have all the variables required for the Area of a Trapezoid Formula?*

Yes, in Steps 1 and 2, I defined all four variables:

\[\purple b_1=3\]

\[\purple b_2=10\]

\[\purple h=4\]

\[\purple A=A\]

When I substitute these values into the Area of a Trapezoid Formula, I get…

\[{\purple A}=\frac{1}{2}({\purple 4})({\purple 3}+{\purple 10})\]

I can simplify the right side of the equation by adding \(\purple 3\) and \(\purple 10\).

\[{\purple A}=\frac{1}{2}({\purple 4})(13)\]

Then I can further simplify it by multiplying \(\frac{1}{2}\) by \(\purple 4\).

\[{\purple A}=2(13)\]

Then I can multiply \(2\) by \(13\).

\[{\purple A}=26\]

The area of the trapezoid is \(26 \; ft^2\).

Find the missing base of the trapezoid if the area of the trapezoid is \(76 \; mm^2\).

*What measurements were given to you?*

The problem told me that the area of the trapezoid is \(76 \; mm^2\).

\[\red A=76\]

The diagram tells me that one of the bases is \(13 \; mm\) and the height of the trapezoid is \(8 \; mm\).

\[\red b_1=13\]

\[\red h=8\]

*Which measurement are you looking for?*

The measurement I want to find is the length of the missing base.

\[\red b_2 = b\]

*Do you have all the variables required for the Area of a Trapezoid Formula?*

Yes, in Steps 1 and 2, I defined all four variables:

\[\red A=76\]

\[\red b_1=13\]

\[\red h=8\]

\[\red b_2 = b\]

When I substitute these values into the Area of a Trapezoid Formula, I get…

\[{\red 76}=\frac{1}{2}({\red 8})({\red 13}+{\red b})\]

I can simplify the right side of the equation by multiplying \(\frac{1}{2}\) by \(\red 8\).

\[{\red 76}=4({\red 13}+{\red b})\]

Then I can divide both sides of the equation by \(4\).

\[19={\red 13}+{\red b}\]

Lastly, I will subtract \(\red 13\) from both sides of the equation to isolate the \(\red b\).

\[6={\red b}\]

The missing base of the trapezoid is \(6 \; mm\) long.

## Area of a Trapezoid Formula Proof

The Area of a Trapezoid Formula works because…

The first part of the Area of a Trapezoid Formula is \(\yellow \frac{1}{2}h\). We can visualize this on a diagram by cutting the trapezoid in along the dotted yellow line.

If we rotate the top part of the cut trapezoid and line up the slanted sides, we end up with a parallelogram.

The base of the parallelogram is \(\red b_1+b_2\) and the height of the parallelogram is \(\yellow \frac{1}{2}h\).

The Area of a Parallelogram Formula is \(A=bh\) where \(b\) is the base of the parallelogram and \(h\) is the height.

If we plug \(\red b_1+b_2\) and \(\yellow \frac{1}{2}h\) into the Area of a Parallelogram Formula, we get \(A=({\red b_1+b_2})({\yellow \frac{1}{2}h})\). Multiplication is commutative so we can rearrange the formula to get the Area of a Trapezoid Formula \(A={\yellow \frac{1}{2}h}({\red b_1+b_2})\).