Area of Trapezoids Formula

Area of trapezoids are the area of a trapezoid is half the sum of its two parallel bases multiplied by the height: A = 1/2(b_1 + b_2)h.

The Formula

A=12(b1+b2)×hA = \frac{1}{2}(b_1 + b_2) \times h

When to use: Two identical trapezoids fit together to form a parallelogram. The trapezoid is half of that parallelogram.

Quick Example

A trapezoid with bases 6 cm and 10 cm, and height 4 cm: A=12(6+10)×4=32A = \frac{1}{2}(6 + 10) \times 4 = 32 cm2^2.

Notation

b1,b2b_1, b_2 = parallel bases, hh = perpendicular height

What This Formula Means

The area of a trapezoid is half the sum of its two parallel bases multiplied by the height: A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h.

Two identical trapezoids fit together to form a parallelogram. The trapezoid is half of that parallelogram.

Worked Examples

Example 1

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A trapezoidal deck has bases 1212 ft and 2020 ft and height 99 ft. Find its area.

Answer

144 ft²

First step

1
A=12(12+20)(9)=12(32)(9)A=\tfrac{1}{2}(12+20)(9)=\tfrac{1}{2}(32)(9).

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Example 2

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A trapezoid has vertices (0,0)(0,0), (10,0)(10,0), (7,5)(7,5), (2,5)(2,5). Find its area.

Example 3

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An isosceles trapezoid has parallel sides 66 and 1414 and legs of 55. Find its area.

Common Mistakes

  • Multiplying only one base by the height - average the two parallel bases first: 12(b1+b2)\frac{1}{2}(b_1+b_2).
  • Using a slanted leg as the height - the height is the perpendicular distance between the two parallel bases.
  • Forgetting the 12\frac{1}{2} after adding the bases - the formula averages the bases, so the half is essential.

Why This Formula Matters

It generalizes the rectangle and parallelogram (where the two bases are equal) and is the workhorse for composite-figure and under-a-graph area. The key insight — that you average the bases because the figure widens or narrows — is what later powers the trapezoidal estimate of area in statistics and calculus. Recognizing it by "Do I have two parallel bases of different lengths and the perpendicular height between them?" — rather than by familiar numbers — is what lets a student tell it apart from area of a parallelogram and area of a triangle and perimeter of a trapezoid in a mixed problem set.

Frequently Asked Questions

What is the Area of Trapezoids formula?

The area of a trapezoid is half the sum of its two parallel bases multiplied by the height: A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h.

How do you use the Area of Trapezoids formula?

Two identical trapezoids fit together to form a parallelogram. The trapezoid is half of that parallelogram.

What do the symbols mean in the Area of Trapezoids formula?

b1,b2b_1, b_2 = parallel bases, hh = perpendicular height

Why is the Area of Trapezoids formula important in Math?

It generalizes the rectangle and parallelogram (where the two bases are equal) and is the workhorse for composite-figure and under-a-graph area. The key insight — that you average the bases because the figure widens or narrows — is what later powers the trapezoidal estimate of area in statistics and calculus. Recognizing it by "Do I have two parallel bases of different lengths and the perpendicular height between them?" — rather than by familiar numbers — is what lets a student tell it apart from area of a parallelogram and area of a triangle and perimeter of a trapezoid in a mixed problem set.

What do students get wrong about Area of Trapezoids?

The procedure for area of trapezoids is the easy part; the trap is multiplying only one base by the height. Asking "Do I have two parallel bases of different lengths and the perpendicular height between them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Area of Trapezoids formula?

Before studying the Area of Trapezoids formula, you should understand: area of parallelograms, area of triangles.

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