How to Find the Area of a Trapezoid: Formula & Example Problems

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Calculate the area of any trapezoid and find missing dimensions
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Trapezoids are a unique shape found in popcorn buckets, lampshades, and…math problems. If you’ve been tasked to find the area of a trapezoid, the solution is actually fairly simple! You just need the right formula (A = ½ (a + b) h) and to find all of the dimensions of the shape—if you don’t have them already. Follow the simple, student-friendly steps and example problems below to easily solve for the area of any trapezoid.

Trapezoid Area Formula Overview

The formula A = ½ (a + b) h is used to find the area of a trapezoid, where A = area, a = top base length, b = bottom base length, and h = height. To calculate the shape’s area, plug these values into the formula and solve. Add square units (e.g., in 2 ) to your final answer.

Section 1 of 6:

Area of a Trapezoid Formula

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  1. In this formula, A is the area of the trapezoid, a and are the bases (or parallel sides) of the shape, and h is the height (or the perpendicular distance between the bases. Here’s an example of how to use this formula with all dimensions given: [1]
    • Example Problem: Find the area of a trapezoid whose parallel sides are 32 cm and 12 cm, respectively, and whose height is 5 cm. Solution:
      • The known dimensions are a = 32 cm, b = 12 cm, and h = 5 cm.
      • Plug these values into the formula: A = ½ (32 + 12) × (5) = ½ (44) × (5)
      • Solve to get 110 .
      • Add the appropriate units to get 110 cm 2 .
    • Note: Trapezoids that have equal bases (i.e., a = b ) result in a simplified formula: A = a × h. This is the same formula used to calculate the area of a rectangle .
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Section 2 of 6:

How to Calculate the Area of a Trapezoid

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  1. A trapezoid has two bases—the top and bottom of the shape, or the two non-slanted lines that are parallel to each other. Check if you were given the measurements for each base—these measurements are your a and b values. [2]
    • For example , the trapezoid shown above has values a = 8 cm and b = 13 cm.
    • If you don’t know one of the bases , you can use a ruler to measure it (if the trapezoid shape is up to scale). Otherwise, you’ll need to already know the trapezoid’s perimeter or area and use those formulas to work backwards and find the missing base length.
  2. The height of the trapezoid is the distance between the parallel bases. Check if you were given the measurement for this line—if so, this measurement is your h value. [3]
    • For example , the trapezoid shown above has the value h = 7 cm.
    • Keep in mind that the length of the angled sides, or the legs of the trapezoid, is not the same as the height. The leg length is only the same as the height if the leg is perpendicular to the bases.
    • If you don’t know the height , you can use a ruler to measure it (if the trapezoid shape is up to scale). If you were given the side lengths of the trapezoid but not the height, proceed to this set of steps .
  3. Now that you have all of your measurements, you’ll plug your values into the formula for the area of a trapezoid . Take the base values a and b and add them together to get a sum. Then, multiply the sum by value h to get a final product. Add the appropriate units and take note of your result for use in the next step. [4]
    • For example , take the values a = 8 cm, b = 13 cm, and h = 7 cm. 8 cm + 13 cm = 21 cm. 21 cm x 7 cm = 147 cm 2 .
  4. You can either multiply the product by ½ or divide the product by 2 to get the final area of the trapezoid since the result will be the same. Write and label your final answer in square units. [5]
    • For this example, 147 cm 2 / 2 = 73.5 cm 2 , which is the area (A) of the trapezoid.
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Section 3 of 6:

Calculating Area of a Trapezoid Without Known Height

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  1. Drawing the shapes helps you visualize the area better and helps you find the height of the trapezoid. Draw straight lines down from the corners of the top base so they intersect and form 90-degree angles with the bottom base. The inside of the trapezoid will have 1 rectangle in the middle and 2 triangles on either side that are the same size and have 90-degree angles. [6]
    • This method only works for regular trapezoids.
  2. Subtract the length of the top base from the length of the bottom base to find the amount that’s left over. Divide the amount by 2 to find the length of the triangle’s base. You should now have the length of the base and the hypotenuse of the triangle . [7]
    • For example, if the top base (b 1 ) is 6 cm and the bottom base (b 2 ) is 12 cm, then the base of the triangle is 3 cm (because b = (b 2 - b 1 )/2 and (12 cm - 6 cm)/2 = 6 cm which can be simplified to 6 cm/2 = 3 cm).
  3. Plug the values for the length of the base and the hypotenuse, or the longest side of the triangle, into A 2 + B 2 = C 2 , where A is the base and C is the hypotenuse. Solve the equation for B to find the height of the trapezoid. If the length of the base you found is 3 cm and the length of the hypotenuse is 5 cm, then, in this example: [8]
    • Fill in the variables: (3 cm) 2 + B 2 = (5 cm) 2
    • Simplify the squares: 9 cm +B 2 = 25 cm
    • Subtract 9 cm from each side: B 2 = 16 cm
    • Take the square root of each side: B = 4 cm

    Tip: If you don’t have a perfect square in your equation, then simplify it as much as possible and leave a value with a square root. For example, √32 = √(16)(2) = 4√2.

  4. Put the base lengths and the height into the formula A = ½(b 1 +b 2 )h to find the area of the trapezoid. Simplify the number as much as you can and label it with square units. [9]
    • Write the formula: A = ½(b 1 +b 2 )h
    • Fill in the variables: A = ½(6 cm +12 cm)(4 cm)
    • Simplify the terms: A = ½(18 cm)(4 cm)
    • Multiply the numbers together: A = 36 cm 2 .
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Section 4 of 6:

Sample Questions for Trapezoid Area

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  1. 1
    Example 1: Known Dimensions & Same Units Find the area of a trapezoid with bases of lengths 10 cm and 14 cm and a height of 5 cm. Solution:
    • Write out the values: a = 10, b = 14, h = 5.
    • Add together the bases: 10 + 14 = 24.
    • Multiply this sum by the height: 24 x 5 = 120.
    • Divide this product by 2: 120 / 2 = 60.
    • Add the units to write your final answer: 60 cm 2 .
  2. 2
    Example 2: Known Dimensions & Different Units Find the area of a trapezoid with bases of lengths 5 yards and 126 in and a height of 4 yards. Solution:
    • Write out the values: a = 5 yd, b = 126 in, h = 4 yd.
    • Since all units must be the same, convert 126 in into yards: 1 yd = 36 in. 126 / 36 = 3.5. 126 in = 3.5 yards
    • Rewrite your values with like units: a = 5 yd, b = 3.5 yd, h = 4 yd.
    • Add together the bases: 5 + 3.5 = 8.5.
    • Multiply this sum by the height: 8.5 x 4 = 34.
    • Divide this product by 2: 34 / 2 = 17.
    • Add the units to write your final answer: 17 yd 2 .
  3. 3
    Example 3: Without the Height Dimensions Find the area of a trapezoid with bases of lengths 10 cm and 20 cm, and side lengths of 8 cm each. Solution:
    • Write out the values: a = 10, b = 20, h = X.
    • Split your trapezoid into a rectangle with two triangles on either side.
    • Subtract the length of the longer base by the shorter one: 20 - 10 = 10.
    • Divide the difference by 2: 10 / 2 = 5. This is the length of the base of one of the triangles.
    • Plug this base value and the hypotenuse value (8) into the Pythagorean theorem to solve for the triangle’s height: A 2 + B 2 = C 2 → 5 2 + B 2 = 8 2 .
    • Simplify the squares: 25 + B 2 = 64.
    • Isolate B: B 2 = 39.
    • Take the square root of each side: B = 6.24.
    • Plug B into the trapezoidal area formula and solve: A = ½ (10 + 20) 6.24 = ½ (30) 6.24 = ½ (187.2) = 93.6.
    • Add the units for your final answer: 93.6 cm 2 .
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Section 5 of 6:

What is the area of a trapezoid?

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  1. Put in mathematical terms, the trapezoid’s area is the number of unit squares that can fit inside it. This area is measured in square units, such as cm 2 , in 2 , and ft 2 . It is calculated by using the lengths of each side of the trapezoid, particularly the bases (the two parallel lines) and the height (the perpendicular length between these bases). [10]
    • But what’s a trapezoid? A trapezoid is a four-sided shape with one pair of parallel sides and one pair of non-parallel sides. The internal angles of a trapezoid always sum up to 360°, while the angles on the same side of a leg always sum up to 180º. [11]
      • Examples of trapezoid-shaped objects include flower planters, handbags, popcorn buckets, and lampshades.
Section 6 of 6:

Frequently Asked Questions About Trapezoids

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  1. 1
    How do I find the perimeter of a trapezoid? You’ll need the height, the lengths of both sides, and the base lengths of a trapezoid in order to solve for its perimeter, says math teacher Joseph Meyer. If you have all the necessary measurements, simply add them together using this formula: Perimeter = a + b + c + d.
  2. 2
    How is a trapezoid different from other quadrilaterals? Trapezoids are a type of quadrilateral. However, they’re distinct from other shapes because they have only one pair of parallel sides. As a result, every parallelogram (including rectangles) is a trapezoid, while not all trapezoids are rectangles. [12]
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Practice Worksheet with Problems & Solutions

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  • Question
    How do I find the area if given only the shorter base and height?
    Donagan
    Top Answerer
    You have to know the lengths of both bases (as well as the height) in order to find the area.
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  • Question
    Why do I divide by two?
    Donagan
    Top Answerer
    You're actually finding the average of the two bases first (by adding their lengths and dividing by two) and then multiplying by the height.
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  • Question
    Will this formula work with every trapezoid?
    Community Answer
    Yes. Even though not all trapezoids are the same size, it will still work if you plug the numbers in correctly.
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      • If you know the median of the trapezoid, which is a line that runs parallel to the bases through the middle of the shape, then multiply it by the height to get the area. [13]
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      References

      1. ↑ https://www.nova.edu/tutoring-testing/study-resources/forms/area-and-volume-formulas.pdf
      2. ↑ https://jwilson.coe.uga.edu/emt668/emat6680.2002/bauers/6690%20Instructional%20Unit/5.2%20/5.2%20Areas%20of%20Tpt.html
      3. ↑ https://jwilson.coe.uga.edu/emt668/emat6680.2002/bauers/6690%20Instructional%20Unit/5.2%20/5.2%20Areas%20of%20Tpt.html
      4. ↑ https://psep.tennessee.edu/common-calculations/
      5. ↑ https://psep.tennessee.edu/common-calculations/
      6. ↑ https://jwilson.coe.uga.edu/emt668/emat6680.2002/bauers/6690%20Instructional%20Unit/5.2%20/5.2%20Areas%20of%20Tpt.html
      7. ↑ https://youtu.be/5KmCDSI3n-8?t=87
      8. ↑ https://youtu.be/5KmCDSI3n-8?t=87
      9. ↑ https://www.nova.edu/tutoring-testing/study-resources/forms/area-and-volume-formulas.pdf
      1. ↑ http://www.geom.uiuc.edu/~demo5337/Group3/area6a.html
      2. ↑ https://tasks.illustrativemathematics.org/content-standards/tasks/1504
      3. ↑ https://tasks.illustrativemathematics.org/content-standards/tasks/1504
      4. ↑ https://www.mathopenref.com/trapezoidmedian.html

      About This Article

      Article Summary X

      To find the area of a trapezoid, start by adding together the length of the bases, which are the 2 sides of the trapezoid that are parallel with each other. Then, multiply that number by the height of the trapezoid. Finish by dividing the product by 2 to find the area. For example, if one of the trapezoid's bases is 8 inches long and the other one is 12 inches long, first you'd add those together and get 20 inches. Then, if the trapezoid's height was 10 inches, you'd add that to 20 and get 30. Just divide 30 by 2 to get 15, which is the area of the trapezoid. To learn how to calculate the area of a trapezoid if you only know the sides, scroll down!

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