Half of a rectangle

Why the area of a right triangle is base times height, divided by two.

You already know how to find the area of a rectangle. You multiply the two sides.

This lesson is about what happens when you cut one in half.

Here is a rectangle with a line drawn corner to corner. The grid lets you check the numbers yourself if you want to.

A rectangle cut corner to corner into two triangles. This one can be dragged about, which needs JavaScript switched on.

Drag either marked corner. Watch the two triangles, and watch the numbers.

Two things are worth noticing, and you can check both by dragging:

The two triangles always look the same as each other. Not similar. The same.
Whatever the rectangle’s area is, each triangle gets exactly half of it.

Try to break it. Make the rectangle very wide. Make it very flat. It keeps happening.

Each triangle has a square corner, the one that came from the corner of the rectangle. A triangle with a square corner in it is called a right triangle, and the square corner is the right angle. It is the same 90 degrees you get where a wall meets the floor.

The two sides that meet at the right angle have names too. One is the base and the other is the height. Which is which does not matter, and that is worth pausing on: you can turn the triangle around and swap them, and nothing about it changes.

The long slanted side, the one that came from the diagonal, has a name as well: the hypotenuse. You will not need it today. It is here so it is not a surprise later.

From rectangle to triangle

Start with the rectangle. Its sides are the base and the height, so its area is

A_{\text{rectangle}} = b \times h

The diagonal cut it into two triangles, and you saw that the two are the same as each other. So each one is half of the rectangle:

A_{\text{triangle}} = \frac{b \times h}{2}

That is the whole result. A right triangle’s area is the base times the height, divided by two.

It is worth saying what this is not. It is not “multiply the three sides”. It is not “multiply the two longest sides”. It is the two sides that meet at the right angle, and only those two. The slanted side never enters into it.

You never need to draw the rectangle to use this — once you can pick out the two sides meeting at the right angle, the halving is automatic. That’s the payoff: a shortcut for a construction you now know how to justify from scratch if you forget it.

A triangle with a base of 10 and a height of 6.

The rectangle it came from would be $10 \times 6 = 60$ . So the triangle is half of that:

\frac{10 \times 6}{2} = 30

A triangle with a base of 7 and a height of 3.

Same move. The rectangle would be $21$ , so the triangle is

\frac{7 \times 3}{2} = 10.5

Notice that the answer came out as a half, and that is fine. Halving an odd area is allowed. Area is not obliged to be a whole number.

One to watch out for. A triangle with sides 3, 4 and 5 has a right angle in it, and the two sides that meet at that right angle are the 3 and the 4.

So its area is

\frac{3 \times 4}{2} = 6

not $\frac{4 \times 5}{2} = 10$ . The 5 is the slanted side. It is the longest, which makes it tempting, and it is exactly the one you must not use.

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.