Area of triangle

For the area of a triangle you meet two new things: besides the side, you also need the altitude (height); and at the end you divide by two. It looks unusual at first, but in a moment you will see why.

The formula

First pick one side of the triangle – say, the side we call side a. To that side belongs one altitude, which we call altitude hₐ (read "h-a"). Then:

A = (a · hₐ) / 2

In words: the area of a triangle equals half the product of the side and the altitude that goes with it.

What the altitude of a triangle is

The altitude is the perpendicular distance from the opposite vertex of the triangle down to the side. Imagine side a is the bottom edge of the triangle. The top vertex is somewhere above it. Drop a perpendicular line down – its length is altitude hₐ.

In the picture you see the altitude as a dashed line perpendicular to the side. A small square shows where the right angle is – that means the lines are really perpendicular.

Why we divide by two

Picture a rectangle with sides a and hₐ. Its area is a · hₐ. Now cut that rectangle along a diagonal – you get two equal triangles. Each triangle has exactly half the area of the rectangle.

So: triangle area = half of rectangle area = (a · hₐ) / 2.

Worked example

A triangular sail has a side of 8 m and an altitude of 6 m to that side. What is its area?

Solution: A = (8 · 6) / 2 = 48 / 2 = 24 m². The area is 24 square meters.

Common mistakes

Forgetting to divide by two. Without that step you would get the area of a rectangle, not a triangle. Always remember: divide by two at the end.
Using the wrong altitude. Each side of a triangle has its own altitude. Always use a side and its matching altitude together.
Confusing altitude with a side. The altitude must be perpendicular to a side. A slanted side of the triangle is not the altitude!

Nice numbers in the practice

In the generator the side and altitude are chosen so that their product is always even. That way after dividing by two the result is always a whole number.

Practice

Generator: Area of triangle