Area of a triangle — why it is base × height ÷ 2

On Board 1, the two sliders on the left set the base and the height. The triangle and the parallelogram on the right are both wired to those two sliders, so changing the base or the height reshapes both figures and updates both areas.

A triangle is exactly half a parallelogram

The board starts with a base of 6 and a height of 4. The parallelogram's area is 6 × 4 = 24. The triangle's is 6 × 4 ÷ 2 = 12 — exactly half of 24.

The two figures use the same base and the same height. The only difference in the formula is the ÷ 2 at the end.

Why half

A line across a parallelogram from one corner to the opposite corner is a diagonal, and it splits the parallelogram into two triangles of exactly the same shape and size.

Each of those triangles has the same base and height as the parallelogram. Two identical triangles make up the whole, so one of them is half.

It works the other way round too. Any triangle and a copy of it turned half a turn, fitted together along a side that is not the base, form a parallelogram with the same base and height.

And why is a parallelogram base × height? Cutting a right triangle off one end and moving it to the other end turns the parallelogram into a rectangle. The base and height have not changed, so the area is the rectangle's, base × height. A triangle is half of that: base × height ÷ 2.

Moving the top corner keeps the area

A base and a height do not fix a triangle's shape. With the same base and height, the top corner can sit in different places. Board 2 wires the same base and height into two differently shaped triangles.

The top figure has its corner above a point inside the base. The bottom one is a right triangle, with its corner straight above the left end of the base. With base 6 and height 4, both come to 6 × 4 ÷ 2 = 12.

The right triangle is a 6 by 4 rectangle cut in half along its diagonal: half of 24 is 12.

When the top corner slides along a line parallel to the base, its height above the base stays the same, so the area stays the same. That holds even when the corner moves past the end of the base. The height is then measured at a right angle from the base line, extended, up to the corner. The board's figure card cannot draw that shape, so it is described here in words only.

The menu at the top of a figure card switches its shape. With the bottom card set to parallelogram, Board 2 shows the same comparison as Board 1.

Height is not the slanted side

The height is the distance from the base measured at a right angle. On the boards, each figure draws its height as a dashed vertical line beside the shape.

The right triangle on Board 2 also shows the length of its slanted side (the hypotenuse): 7.2111 for a base of 6 and a height of 4. The area does not use that length.

A parallelogram's slanted side is also longer than its height. Multiplying by the slanted side instead of the height gives an area that is too big.

Other numbers

The sliders on Boards 1 and 2 go from 1 to 10 in steps of 1.

In every row the triangle is exactly half the parallelogram. With base 5 and height 3 the parallelogram comes to an odd 15, and the triangle is still exactly half: 7.5.

Base 3 and height 8 give a very different shape from the starting 6 and 4, yet the area is the same 12, because base × height is 24 in both cases.

On Board 2 the two triangles always match: 18 each for base 9 and height 4, 15 each for base 10 and height 3.

What this formula needs

Base × height ÷ 2 works when a base and its height are known. With only the lengths of the three sides, it cannot be used directly: either a height is worked out first, or the area comes from another formula that uses the three sides (Heron's formula).

Read next — Unit conversion — length, area and time It converts length, area and time into other units, and shows why 1 m² is 10,000 cm²: both sides are multiplied by 100.

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