How to find the area and perimeter of a triangle

The area of any triangle can be found with one formula, half the height times the base. The perimeter is even simpler: you add all the side lengths together. The one thing people get wrong is the height, so that's where we'll spend most of the time.

When do I use this?

Whenever you're asked how much space a triangle covers (the area) or how long the way round it is (the perimeter). The area formula is the same for every type of triangle, right-angled, isosceles, equilateral, all of them.

The formula

$$A = \frac{1}{2} \times h \times g$$

Here $h$ is the height of the triangle and $g$ is the base.

Look at the small square where the height meets the base. It marks a right angle. The height is the straight line from a corner down onto the opposite side, and it has to stand perpendicular on that side. It is not one of the slanted sides.

The procedure

1. Pick a side to be the base $g$.
2. Find the height $h$, the distance from the opposite corner straight down to the base, at a right angle.
3. Multiply the height by the base.
4. Take half of it.

Worked examples

A triangle with base 6 and height 4

$$A = \frac{1}{2} \times 4 \times 6 = 12$$

So the area is 12. If the lengths were in centimetres, the area is 12 square centimetres. Area always comes in square units, which is explained in units: metres, square metres and cubic metres.

A triangle with base 10 and height 3

$$A = \frac{1}{2} \times 3 \times 10 = 15$$

Notice that the third side never came into it. We didn't need it.

The perimeter

To find the perimeter we add all the side lengths together. A triangle with sides 3, 4 and 5 has the perimeter

$$3 + 4 + 5 = 12$$

Here we do need every side, because the perimeter is the whole way round.

Common mistakes

• "I need all three sides to find the area." You don't. Height and base are enough. The third side is only needed for the perimeter.
• "The height is one of the sides." Only if that side stands at a right angle to the base, which happens in a right-angled triangle. Otherwise the height is the separate line drawn straight down from the corner, like the dashed line in the figure.
• "Area and perimeter are the same kind of thing." They aren't. The perimeter is a length, the distance round the outside. The area is how much is inside, and it comes in square units.

Related

Everything about the types of triangle is in Triangles: types, angles and properties. If the question is about angles instead of lengths, see the angles in a triangle add up to 180 degrees. And if you're given one triangle and need a side in another one of the same shape, that's similar triangles.