✂️ Cut, Slide, Find: Area of a Parallelogram and a Triangle

How do we find the area of a parallelogram? A ruler is not enough, because its sides are slanted. But with a pair of scissors and a little sliding, it is easy! Let’s see.

First, two new words. The bottom side is the base. The perpendicular distance between the base and the top side is the height. The slanted side is not the height, however long it is.

Now let’s draw the height down from a corner to the base, and cut right along that line with scissors to take off a small triangle.

We slide the triangle to the other side. It fits exactly! The parallelogram has turned into a rectangle. Nothing was added and nothing was thrown away. The area stayed the same.

The area of a rectangle is the number of 1 square centimetre squares that fit inside it. The base is 8 cm and the height is 4 cm: 4 rows, 8 squares in each row, 32 squares. The area is 32 cm². So the area of a parallelogram is base times height.

Now let’s look at a triangle. If we cut a parallelogram along its diagonal, we get two equal triangles. If we turn one triangle half a turn and put it next to the other, we make a parallelogram again. This parallelogram is 6 times 4, which is 24 square centimetres.

The parallelogram has an area of 24, and the triangle is half of it: 24 divided by 2 is 12 cm². So the area of a triangle = base × height ÷ 2.

Let’s slide the top corner of the triangle left and right along the top line. The shape changes, but the base and the height stay the same, so the area is always 12 square centimetres. The height is the perpendicular distance from the top corner down to the base.

Summary: we cut a parallelogram, slide a piece and make a rectangle, so the area is base times height. The height drops straight down, and the slanted side is not the height. Two equal triangles make a parallelogram, so the area of a triangle is base times height divided by two. The unit is the square centimetre. Now try it in Area Scissors!