Area of Rectangles and Triangles

Area measures the space covered inside a shape, and the honest way to picture it is counting squares. This rectangle is 8 centimetres across and 5 down, so it holds 5 rows of 8 little one-centimetre squares, which is 40 squares altogether. Each of those squares is one square centimetre, so the area is 40 square centimetres. You would not want to count them one at a time forever, and you do not have to, because 5 rows of 8 is just 8 times 5 — and that is where the formula comes from. Keep area and perimeter apart in your head: perimeter is the trip around the edge, area is the covering inside, and the unit tells you which one you have found.

Because a rectangle holds rows of squares, its area is simply the length times the width. For one measuring 9 centimetres by 6, that is 9 times 6, which is 54 square centimetres — six rows of nine squares. It does not matter which side you call the length and which the width, as long as the two sides you multiply meet at a corner. A square is just a rectangle whose sides are equal, so its area is the side times itself: a square of side 7 has an area of 49 square centimetres. One habit worth building now is reading the question twice. This same 9 by 6 rectangle has a perimeter of 30 centimetres and an area of 54 square centimetres, and answering the wrong one scores nothing.

Now for triangles, and the trick is to draw a box around one. Take a triangle with a base of 8 centimetres and a height of 6. Enclose it in a rectangle 8 by 6, then drop the height line down from the top corner. That line splits the box into two smaller rectangles, and the triangle takes exactly half of each of them — so the triangle is exactly half the whole box. The box has an area of 8 times 6, which is 48 square centimetres, so the triangle is half of that, or 24 square centimetres. That is the formula: a half times base times height. Multiply base by height first and halve at the end, because the arithmetic is easier that way.

Here is the detail that decides most triangle marks. The height in the formula is the perpendicular height, meaning the straight-up distance from the base measured at a right angle to it — not the length of a slanted side. In the left triangle the base is 9 and the perpendicular height, drawn as a dashed line inside the shape, is 4, so the area is a half times 9 times 4, which is 18 square centimetres. That 5 on the slanted side is there to tempt you, and using it would give 22.5, which is wrong. A right-angled triangle is friendlier: its two short sides already meet at a right angle, so they are the base and height. Six and eight give an area of 24 square centimetres.

Two last skills. First, composite shapes. This one is a rectangle with a triangle sitting on top, so cut it along the dashed line and do the pieces separately. The rectangle is 10 by 4, giving 40 square centimetres, and the triangle has base 10 and height 3, giving a half of 30, which is 15. Add them for a total of 55 square centimetres. Second, working backwards. If a triangle has an area of 24 square centimetres and a base of 8, write the formula, substitute to get 24 equals a half times 8 times h, simplify the known part to 24 equals 4h, and divide to get a height of 6 centimetres. Then check forwards. And remember: cutting is fine for area, never for perimeter.