Area of triangles and trapezoids

The area of a flat shape is the amount of space inside it. It tells you how much paint covers a wall, or how many square tiles fill a floor. Rectangles are easy: you multiply the length by the width. Triangles and trapezoids look harder, but each one can be built from a rectangle. People have used these rules for a very long time. An Egyptian scroll called the Rhind papyrus, written about 3,500 years ago, already shows how to find the area of a triangular field and of a field shaped like a trapezoid.

Base and height

Every triangle has a base and a height. The base is one side, usually the one at the bottom. The height is the distance from the base straight up to the top corner. "Straight up" means at a right angle to the base, like a wall standing on a floor. The height is not the slanted side. It is the shortest way from the top corner down to the line of the base.

In a right triangle this is very easy. The two short sides already meet at a right angle, so one of them is the base and the other one is the height. The long slanted side, the hypotenuse, is not used for the area.

Why a triangle is half a rectangle

Draw a rectangle that is 10 wide and 6 tall. Its area is 10 × 6 = 60. Now draw one diagonal, from one corner to the opposite corner. The diagonal cuts the rectangle into two right triangles. The two triangles are exactly the same size, so each one has half of the area: 30.

This works for every triangle, not only right ones. Draw the rectangle around the triangle, with the same base and the same height. The height line from the top corner cuts the rectangle into two smaller rectangles. In each small rectangle, the triangle covers exactly half. So the whole triangle covers half of the big rectangle. Sometimes the top corner leans out past the end of the base. Then the height falls outside the triangle and the picture is a little different, but the rule is still true.

1. Find the base and the height.
2. Multiply them. This is the area of the rectangle around the triangle.
3. Divide by 2. This is the area of the triangle.

What a trapezoid is

A trapezoid is a shape with four straight sides, where two of the sides are parallel. Parallel lines go in the same direction and never meet, like the two rails of a railway. The two parallel sides are usually drawn at the top and at the bottom. They have different lengths. The height of a trapezoid is the distance between the two parallel sides, measured straight across, at a right angle.

The area of a trapezoid

Here is the teacher's way to see the rule. Draw a diagonal inside the trapezoid. It cuts the trapezoid into two triangles. One triangle has the top side as its base. The other triangle has the bottom side as its base. Both triangles have the same height as the trapezoid.

For the trapezoid above, the first triangle has area 1/2 × 3 × 5 = 7.512 × 3 × 5 = 7.5. The second has area 1/2 × 9 × 5 = 22.512 × 9 × 5 = 22.5. Together that is 7.5 + 22.5 = 30.

Look at what we did: we took half of the top, half of the bottom, and multiplied by the height. That is the same as adding the two parallel sides, dividing by 2, and multiplying by the height. Adding two numbers and dividing by 2 gives their average, the number exactly in the middle. The average of 3 and 9 is 6. So the trapezoid has the same area as a rectangle 6 wide and 5 tall: 30.

1. Add the two parallel sides.
2. Divide by 2. This is the average width.
3. Multiply by the height.

Does the answer make sense?

• If the top side of a trapezoid shrinks to 0, the trapezoid becomes a triangle. The rule then gives (0 + b)/2 × h0 + b2 × h, which is half of base times height: the triangle rule.
• If the top and the bottom are the same length, the trapezoid becomes a rectangle. The average of two equal numbers is that number, so the rule gives width times height.
• A triangle's area is always half of the rectangle around it. A trapezoid's area is always between the two rectangles made from its short side and its long side.