Area of a triangle

Draw a rectangle, draw one of its diagonals (the line from one corner to the opposite corner), and cut along it. You get two triangles, and when you place one on the other they overlap exactly. So a diagonal cuts a rectangle into two triangles of equal area, and each triangle is half of the rectangle.

On grid paper, the book draws rectangle ABCD: the bottom side AB is 5 squares long and the side AD is 4 squares tall, so its area is 5 × 4 = 20 sq units. The blue triangle BAD has corners A, B and D; its side BD is a diagonal. So its area is half of 20, which is 10 sq units.

The red triangle ABE is different. Its side is the whole bottom AB, and its third corner E is on the top side DC, 3 squares from D and 2 squares from C. It looks very different from BAD, but its area is the same. Drop a line straight down from E to F on AB. This splits ABE into triangle AEF and triangle BEF.

• AEF is half of rectangle AFED (3 by 4, so 12 sq units), which is 6.
• BEF is half of rectangle BFEC (2 by 4, so 8 sq units), which is 4.

Together, 6 + 4 = 10 sq units: half of ABCD.

So: any triangle with one side along a side of a rectangle, and its third corner on the opposite side, has half the area of the rectangle. The side along the rectangle is called the triangle's base, and the rectangle's other side is its height; later you will write this as ½ × base × height. You can use this to find the area of a figure on grid paper: split it into rectangles and triangles, find each area, and add.