How to find the area of a trapezoid
How to find the area of a trapezoid

In this explainer, we will learn how to find the area of a trapezoid using a formula and apply it in finding the area in real life.

We will begin by discussing the different types of trapezia that exist. A trapezoid is a quadrilateral that has exactly one pair of parallel sides. We refer to these sides as the bases and often label their lengths using the letters and . The perpendicular distance between the two bases is called the height of the trapezoid and is usually denoted by . The other two sides of the trapezoid (the nonparallel opposite sides) are known as the legs.

An isosceles trapezoid is a trapezoid in which the legs are of equal length. All isosceles trapezia have a line of symmetry through the midpoints of their bases.

A right trapezoid is a trapezoid in which one of the legs is perpendicular to the two parallel bases.

We will now consider how to find the area of a trapezoid in which we denote the height by and the lengths of the bases by and .

Suppose we draw in a diagonal of the trapezoid, connecting two opposite vertices. This divides the trapezoid into two triangles, as shown in the figure below.

The area of each triangle can be found using the formula

The perpendicular height of each triangle is . The upper triangle has a base of length units, and the lower triangle has a base of length units. Hence, the area of the trapezoid is given by

We note that is the sum of the lengths of the trapezoid’s parallel bases. Informally, we can think of the area of a trapezoid as “half the sum of the parallel bases, multiplied by the height.”

Formula: The Area of a Trapezoid

The area of a trapezoid is equal to half the sum of the lengths of the parallel bases multiplied by the height.

For a trapezoid with height and bases of lengths and , the area is given by

In our first example, we will demonstrate how to apply this formula to find the area of a trapezoid given its height and the lengths of its bases.

Example 1: Finding the Area of a Trapezoid

The parallel sides of a trapezoid have lengths 82 and 70. If the height is 100, what is the trapezoid’s area?

Answer

We recall that the area of a trapezoid is given by where and represent the lengths of the bases, or parallel sides, of the trapezoid and represents its perpendicular height. We have been given each of these lengths in the question.

Substituting , , and and then evaluating gives

Hence, the area of the trapezoid is 7 600 square units.

We have now seen an example of how to calculate the area of a trapezoid given its height and the lengths of its parallel sides. We now consider a problem that is essentially the reverse of this: calculating the height of a trapezoid given its area and the lengths of its bases.

Example 2: Finding the Height of a Trapezoid given Its Area

This trapezoid’s area is 30 000 yd2. What is its height?

Answer

We begin by recalling that the area of a trapezoid can be calculated by multiplying half the sum of the lengths of the parallel bases by the perpendicular height.

From the figure, we identify that the parallel bases of the trapezoid have lengths 80 yd and 295 yd. We have also been given the length of one leg of the trapezoid (232 yd), but this is not relevant to our calculation, as it is not the perpendicular height of the trapezoid.

We can use the given area of the trapezoid and the lengths of the two parallel bases to form an equation where represents the unknown height of the trapezoid:

We now solve this equation to determine the value of . Simplifying the left-hand side gives

Multiplying both sides of the equation by (the multiplicative inverse of ) gives

Hence, the height of the trapezoid is 160 yd.

In the example we have just seen, we were given more information than we needed in the figure. The length of the leg of the trapezoid was not required in order to calculate its area. Understanding the measurements needed to apply a particular formula and being able to select the relevant information from a diagram or worded description are important skills when answering geometric problems.

In our next example, we will consider how to find the length of one of the parallel sides of a trapezoid given its area, its height, and the length of the other parallel side.

Example 3: Finding the Length of a Base of a Trapezoid given Its Area

A trapezoid has area 1 760 and the distance between its parallel sides is 40. If one parallel side is 39, what is the other side?

Answer

We recall that the area of a trapezoid with parallel sides (or bases) of lengths and units and height units is given by

We are given that this trapezoid has an area of 1 760 square units. The distance between the parallel sides, which is another way of saying the height of the trapezoid, is 40 units and the length of one parallel side, or base, of the trapezoid is 39 units. Substituting each of these values into the formula above gives an equation we can solve to determine the length of the other parallel side:

We begin by simplifying the right-hand side of the equation by canceling out a factor of 2:

Dividing both sides of the equation by 20 gives

Finally, subtracting 39 from each side of the equation gives

Hence, the length of the other parallel side (or base) of the trapezoid is 49 units.

We now consider an alternative way of specifying the formula for the area of a trapezoid. Informally, we said this formula can be thought of as “half the sum of the parallel bases, multiplied by the height.” In fact, “half the sum of the parallel bases” has a geometric significance, which we define below.

Definition: The Middle Base of a Trapezoid

The middle base of a trapezoid is the line segment whose endpoints are the midpoints of the legs of the trapezoid. The middle base of a trapezoid is parallel to the trapezoid’s two bases.

The length of the middle base of a trapezoid, , is the arithmetic mean of the lengths of the bases and :

In other words, the length of the middle base of a trapezoid is equal to “half the sum of the lengths of the parallel bases.” The formula for the area of a trapezoid can, therefore, equivalently be expressed as

Formula: The Area of a Trapezoid Using Its Middle Base Length

The area of a trapezoid is equal to the length of its middle base multiplied by its height.

For a trapezoid with height and middle base of length , the area is given by

We now consider two examples in which we use this version of the area formula to solve two problems relating to the area of a trapezoid.

Example 4: Using the Middle Base to Find the Area of a Trapezoid

Find the area of the shown trapezoid.

Answer

Upon inspection of the diagram, we observe that divides each of the legs of the trapezoid, and , into two segments of equal length. Hence, connects the midpoints of the trapezoid’s legs and is, therefore, the middle base of the trapezoid.

We recall that the area of a trapezoid can be calculated from the length of its middle base and its height using the formula

Substituting 19 mm for the length of the middle base and 8 mm for the height of the trapezoid gives

Example 5: Calculating the Middle Base Length of a Trapezoid given Its Area and Height

Find the middle base length of a trapezoid that has an area of 28 cm2 and a height of 4 cm.

Answer

We recall that the area of a trapezoid can be calculated using the formula

We are given the area and height of the trapezoid, and so we can form an equation. Substituting 28 for the area and 4 for the height gives

To find the length of the middle base, we divide both sides of the equation by 4:

The middle base length of the given trapezoid is 7 cm.

The methods we have developed in this explainer can also be applied to real-world problems involving trapezoids. We now consider one final example related to the measurements of a farmer’s fields, one of which is in the shape of a trapezoid.

Example 6: Using the Areas of Trapezoids to Solve a Real-World Problem

A farmer owns two fields of equal area: one in the shape of a rhombus and one in the shape of a trapezoid, as shown in the figure. Calculate the length of the middle base of the trapezoidal field.

Answer

We begin by considering the field on the left in the figure. This is the field in the shape of a rhombus because its four side lengths are all equal. We are given the lengths of the rhombus’s two diagonals: they are 100 m and 90 m. We then recall that the area of a rhombus is given by where and represent the lengths of its diagonals. Hence, we have

Next, we consider the field on the right in the figure. This field is in the shape of a trapezoid because it is a quadrilateral with one pair of parallel sides. We are given the height of the trapezoid (25 m) and we now know that its area, which is the same as the area of the other field, is 4 500 m2.

As we wish to calculate the length of the middle base of the trapezoid, we recall the formula for the area of a trapezoid that involves this measure:

Substituting the known area of the trapezoid (4 500 m2) and the known height (25 m) gives an equation we can solve to determine the length of the middle base:

Dividing both sides of this equation by 25 gives

The length of the middle base of the field in the shape of a trapezoid is 180 m.

Let us finish by recapping some key points.

Key Points

  • The area of a trapezoid with height and parallel bases of lengths and is given by
  • We may think of this informally as
  • The middle base of a trapezoid is the line segment whose endpoints are the midpoints of the two legs of the trapezoid.
  • The length of the middle base of a trapezoid, , is the arithmetic mean of the lengths of the bases:
  • The area of a trapezoid is equal to the length of its middle base multiplied by its height:

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