Using the Properties of Trapezoids to Solve Problems

Learning Outcomes

  • Use properties of trapezoids

A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[/latex], and the length of the bigger base [latex]B[/latex]. The height, [latex]h[/latex], of a trapezoid is the distance between the two bases as shown in the image below.

A trapezoid has a larger base, [latex]B[/latex], and a smaller base, [latex]b[/latex]. The height [latex]h[/latex] is the distance between the bases.

A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.

The formula for the area of a trapezoid is:

[latex]{\text{Area}}_{\text{trapezoid}}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex]

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See the image below.

Splitting a trapezoid into two triangles may help you understand the formula for its area.

An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.
The height of the trapezoid is also the height of each of the two triangles. See the image below.

An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. There is an arrow pointing to a second trapezoid. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.
The formula for the area of a trapezoid is

This image shows the formula for the area of a trapezoid and says “area of trapezoid equals one-half h times smaller base b plus larger base B).
If we distribute, we get,

The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.

Properties of Trapezoids

  • A trapezoid has four sides.
  • Two of its sides are parallel and two sides are not.
  • The area, [latex]A[/latex], of a trapezoid is [latex]\text{A}=\Large\frac{1}{2}\normalsize h\left(b+B\right)[/latex] .

 

example

Find the area of a trapezoid whose height is [latex]6[/latex] inches and whose bases are [latex]14[/latex] and [latex]11[/latex] inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information. A trapezoid of base 11 and 14 inches and height of 6 inches.
Step 2. Identify what you are looking for. the area of the trapezoid
Step 3. Name. Choose a variable to represent it. Let [latex]A=\text{the area}[/latex]
Step 4.Translate.

Write the appropriate formula.

Substitute.

 Plugging in 11 and 14 for bases and 6 for height the equation area A equals one half times height times base one plus base two becomes area equals one half times 6 times 11 plus 14.
Step 5. Solve the equation. [latex]A={\Large\frac{1}{2}}\normalsize\cdot 6(25)[/latex]

[latex]A=3(25)[/latex]

[latex]A=75[/latex] square inches
Step 6. Check: Is this answer reasonable?  [latex]\checkmark[/latex]  see reasoning below

If we draw a rectangle around the trapezoid that has the same big base [latex]B[/latex] and a height [latex]h[/latex], its area should be greater than that of the trapezoid.
If we draw a rectangle inside the trapezoid that has the same little base [latex]b[/latex] and a height [latex]h[/latex], its area should be smaller than that of the trapezoid.

A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.
The area of the larger rectangle is [latex]84[/latex] square inches and the area of the smaller rectangle is [latex]66[/latex] square inches. So it makes sense that the area of the trapezoid is between [latex]84[/latex] and [latex]66[/latex] square inches

Step 7. Answer the question. The area of the trapezoid is [latex]75[/latex] square inches.

 

try it

In the next video we show another example of how to use the formula to find the area of a trapezoid given the lengths of it’s height and bases.

example

Find the area of a trapezoid whose height is [latex]5[/latex] feet and whose bases are [latex]10.3[/latex] and [latex]13.7[/latex] feet.

 

try it

 

example

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[/latex] yards and the bases are [latex]8.2[/latex] and [latex]5.6[/latex] yards. How many square yards will be available to plant?

 

try it