Area of a Triangle

Start with a rectangle

The area of a rectangle is its length times its width. A rectangle 6 across and 4 up holds 4 rows of 6 unit squares, so its area is 6 × 4 = 24 squares. The area of a triangle comes from the rectangle drawn around it.

The 6 by 4 rectangle holds 24 unit squares.

Any triangle in the box

Draw a triangle whose base is the bottom edge of the rectangle and whose top corner touches the top edge. The top corner can be anywhere along the top edge, so this can be any triangle at all, not only a neat one.

The base of the triangle is 6. Its height is the distance straight up from the base to the top corner, which is 4, the height of the rectangle.

The triangle stands on the bottom edge, and its top corner is 2 squares along the top edge.

Two rectangles, and half of each

Drop a line straight down from the top corner to the base, at a right angle to it. The line splits the big rectangle into two smaller rectangles, one on each side: here a 2 by 4 rectangle and a 4 by 4 rectangle.

Each sloping side of the triangle runs from one corner of a small rectangle to the opposite corner, so it cuts that rectangle exactly in half. The triangle takes one half of each. Half of 2 × 4 = 8 is 4, and half of 4 × 4 = 16 is 8, so the triangle’s area is 4 + 8 = 12, which is half of 24.

The line from the top corner splits the box into a 2 by 4 rectangle and a 4 by 4 rectangle.

Drag the top corner along the top edge. The two small rectangles change size, but the triangle always takes half of each, so the two halves always add up to half of 24, which is 12.

Half the base times the height

Wherever the top corner sits, the triangle is half of the rectangle that boxes it in. The rectangle’s area is base × height, so the triangle’s area is half of that: area of a triangle = ½ × base × height.

A triangle with a base of 10 cm and a height of 7 cm has an area of ½ × 10 × 7 = 35 cm². The rectangle around it would be 10 × 7 = 70 cm², and 35 is half of 70.

The height is straight up

The height is always measured at a right angle to the base. It is not the length of a sloping side. Slide the top corner farther along and the sloping sides grow longer, but the height stays 4, so the area stays ½ × 6 × 4 = 12.

The top corner can even go past the end of the base. Then the line straight down from it lands outside the triangle, on the base line made longer, and the area is still 12. To see why, cut the triangle into thin strips that run along the base. Sliding the top corner slides each strip sideways, but no strip gets longer or shorter, so together they cover the same area.

Drag the top corner along the dashed line. The base stays 6 and the height stays 4, so the area stays 12, while the two sloping sides and the perimeter grow.

Multiply, then halve

Two mistakes are common. 6 × 4 = 24 is the area of the whole rectangle, but the triangle fills only half of it, so its area is 12. And 6 + 4 = 10 adds the base to the height: area counts squares, so multiply the base by the height, then halve.