Math Core

You already know that the area of a rectangle is length times width. But many shapes are slanted or pointed, like a ramp seen from the side, a slice of pizza or a sail. The good news: with one clever cut, parallelograms and triangles turn into rectangles, so you can find their areas with formulas you almost already know.

Base and height

Every parallelogram and triangle has a base and a height.

Definition

Base and height

The base is any side you choose to be the "bottom." The height is the distance straight across from the base to the opposite side (or opposite corner), measured at a right angle to the base.

The height is not usually a side of the shape. Think of it as how tall the shape stands if you set it on its base. In the pictures below, the height is drawn as a vertical line segment inside the shape.

Area of a parallelogram

A parallelogram with a base of 6 cm. The vertical segment shows its height, 4 cm.

Imagine cutting off the small triangle on the left side of this parallelogram, along the height line. Now slide that triangle all the way to the right end. It fits perfectly against the slanted right side, and the shape becomes a rectangle that is 6 cm long and 4 cm tall.

Nothing was added or thrown away, so the area did not change. The rectangle's area is $6 \times 4 = 24$ square centimeters, and so is the parallelogram's.

Area of a parallelogram

A = b \times h

$b$ is the length of the base and $h$ is the height. Use the height, not the slanted side.

Worked example: A parallelogram

A parallelogram has a base of 12 inches and a height of 7 inches. Its slanted sides are 9 inches long. Find its area.

The slanted side does not matter for area. Use the base and the height:

A = 12 \times 7 = 84

The area is 84 square inches.

Area of a triangle

A triangle with a base of 8 inches. The vertical segment from the top corner shows its height, 5 inches.

Now take two copies of any triangle. Flip one of them upside down and set it next to the other. Together they always form a parallelogram with the same base and the same height as the triangle.

So one triangle is exactly half of a parallelogram. The parallelogram's area is $b \times h$ , so the triangle's area is half of that. For the triangle above, the area is $\dfrac{1}{2} \times 8 \times 5 = 20$ square inches.

Area of a triangle

A = \frac{1}{2} \times b \times h

Multiply the base by the height, then take half. You can take half at any step, whichever makes the arithmetic easier.

Worked example: A triangle with a decimal height

A triangle has a base of 10 meters and a height of 6.5 meters. Find its area.

A = \frac{1}{2} \times 10 \times 6.5 = 5 \times 6.5 = 32.5

Taking half of 10 first made this easy. The area is 32.5 square meters.

When the height is outside the triangle

In a triangle with a wide (obtuse) angle, the height can fall outside the triangle. You still measure it the same way: straight up from the line the base sits on to the top corner.

The base is the bottom side, 4 ft. The height is the vertical segment on the right, 3 ft, which lands outside the triangle.

Worked example: An obtuse triangle

Find the area of the triangle above.

The base is 4 feet and the height is 3 feet, even though the height is outside the shape.

A = \frac{1}{2} \times 4 \times 3 = 6

The area is 6 square feet.

Common mistake

Don't multiply by a slanted side. The height must meet the base at a right angle. If a picture gives you three numbers, find the two that make a right angle: the base and the height. The slanted side is extra information.

Working backward

If you know the area and one measurement, you can find the other.

Worked example: A missing base

A triangle has an area of 36 square centimeters and a height of 8 centimeters. How long is its base?

36 = \frac{1}{2} \times b \times 8

Half of 8 is 4, so $36 = 4 \times b$ . Since $36 \div 4 = 9$ , the base is 9 centimeters.

Check: $\dfrac{1}{2} \times 9 \times 8 = 36$ . ✓

Tip

A right triangle is easy: its two short sides (the legs) meet at a right angle, so one leg is the base and the other is the height. The longest side is never used for area.