Math Core

Floor plans, garden beds and park maps are rarely perfect rectangles. The trick for finding the area of an odd shape is to break it into pieces you already know: rectangles, triangles and parallelograms. In this lesson you will use that trick to get a formula for trapezoids and to find the area of almost any polygon.

Area of a trapezoid

A trapezoid is a four-sided shape with exactly one pair of parallel sides. The two parallel sides are called the bases ( $b_1$ and $b_2$ ). The height is the distance straight across between them.

A trapezoid with bases of 4 m (top) and 10 m (bottom). The vertical segment shows the height, 5 m.

Here is a neat way to see the formula. Take two copies of the trapezoid, flip one upside down, and set them side by side. Together they make a parallelogram. Its base is $4 + 10 = 14$ m (one short base plus one long base), and its height is still 5 m.

The parallelogram's area is $14 \times 5 = 70$ square meters. One trapezoid is half of that, so its area is $70 \div 2 = 35$ square meters.

Area of a trapezoid

A = \frac{1}{2} \times (b_1 + b_2) \times h

Add the two bases, multiply by the height, then take half.

You can also think of it as the average of the bases times the height: the average of 4 and 10 is 7, and $7 \times 5 = 35$ .

Worked example: Using the formula

A trapezoid has bases of 6 inches and 12 inches and a height of 5 inches. Find its area.

A = \frac{1}{2} \times (6 + 12) \times 5 = \frac{1}{2} \times 18 \times 5 = 9 \times 5 = 45

The area is 45 square inches.

Common mistake

Add the bases first. The parentheses in $(b_1 + b_2)$ mean the two bases are combined before you multiply. Also make sure you use the two parallel sides as the bases, not the slanted sides.

Composing and decomposing polygons

For other polygons there is no single formula. Instead, you decompose the shape (split it into smaller pieces), find each piece's area, and add them up.

Area of any polygon

Split the polygon into rectangles, triangles or other shapes you know.
Find the area of each piece.
Add the pieces. (Or, if it is easier, find the area of a bigger shape and subtract the part that is missing.)

The side of a shed: a rectangle 8 ft wide and 5 ft tall with a triangle on top. The triangle's height is 3 ft.

Worked example: Adding pieces

Find the area of the shed wall above.

The horizontal line splits it into a rectangle and a triangle.

Rectangle: $8 \times 5 = 40$ square feet.
Triangle: its base is the top of the rectangle, 8 ft, and its height is 3 ft. So its area is $\dfrac{1}{2} \times 8 \times 3 = 12$ square feet.

Total: $40 + 12 = 52$ . The wall has an area of 52 square feet.

Worked example: Subtracting a missing piece

A picture frame is a rectangle 10 inches by 8 inches, with a rectangular opening 6 inches by 4 inches cut out of the middle. What is the area of the frame itself?

Find the whole rectangle, then subtract the hole.

Whole rectangle: $10 \times 8 = 80$ square inches.
Opening: $6 \times 4 = 24$ square inches.

Frame: $80 - 24 = 56$ square inches.

Worked example: A regular hexagon

A regular hexagon can be split into 6 identical triangles that meet at the center. Each triangle has a base of 6 cm and a height of about 5.2 cm. Estimate the hexagon's area.

One triangle: $\dfrac{1}{2} \times 6 \times 5.2 = 3 \times 5.2 = 15.6$ square cm.

Six triangles: $6 \times 15.6 = 93.6$ . The hexagon's area is about 93.6 square centimeters.

Tip

There is often more than one way to split a shape. Try two different ways: if both give the same total, you can be confident your answer is right.

Practice

Practice 1

A trapezoid has bases of 6 cm and 10 cm and a height of 4 cm. What is its area, in square centimeters?