Calculating the Area of Triangles

1. Using the area formula

The area of any triangle is half of the base multiplied by the perpendicular height: $$A=\frac{1}{2}\times b\times h=\frac{b\times h}{2}$$ It does not matter whether the triangle is acute, right-angled or wide and flat, as long as you use the perpendicular height.

1. 1Identify a base and the matching perpendicular height. Check they are at 90° to each other.
2. 2Write the formula and substitute: $A=\frac{1}{2}\times b\times h$.
3. 3Multiply, then halve. Halve first if one of the numbers is even.
4. 4Write the answer with square units (cm², m², km²).

For example, a triangle with $b=12$ cm and $h=7$ cm has $A=\frac{1}{2}\times 12\times 7=6\times 7=42$ cm². Notice that we halved the 12 first because that is easier than working out $12\times 7=84$ and then halving. When the numbers are decimals, the same method works: $A=\frac{1}{2}\times 9.6\times 5=4.8\times 5=24$ m².

▶ Base 10 cm and height 6 cm: the rectangle has 60 cm² and the triangle covers half of it, 30 cm².

2. Any side can be the base

A triangle has three sides and so three possible bases. Each base has its own perpendicular height, the distance from the opposite corner to that base. Surprisingly, the three products $\frac{1}{2}bh$ are all equal. This is because they all measure the same piece of surface.

▶ Each base has its own matching height. Multiply them, halve, and the answer never changes.

In problems, choose the base and height that you are given or that are easiest to see. In a right-angled triangle the two shorter sides (the ones that meet at the right angle) are the obvious choice, because one is the base and the other is the height.

3. Finding a missing base or height; mixed units

Sometimes the area is given and you must find the base or height. Treat the formula like an equation. From $A=\frac{1}{2}bh$ we get $b\times h=2A$, so $$b=\frac{2A}{h}\text{and}h=\frac{2A}{b}$$ Double the area first, then divide by the length you know.

For example, a triangle has area 54 m² and base 9 m. Then $h=\frac{2\times 54}{9}=\frac{108}{9}=12$ m. Check by going forwards: $\frac{1}{2}\times 9\times 12=54$ ✓.

Units must match. If the base is 1 m 20 cm and the height is 90 cm, change both to the same unit before you multiply. In cm: base = 120 cm and height = 90 cm, so $A=\frac{1}{2}\times 120\times 90=5,400$ cm². If you need the answer in m², divide by 10,000 (because 1 m² = 100 cm × 100 cm = 10,000 cm²): 5,400 ÷ 10,000 = 0.54 m².

4. Shapes made of triangles, and real-life problems

Many real shapes are compound shapes: a house wall with a triangular gable on top, a rectangular plot with a corner cut off, a kite made of two triangles. To find their area, split the shape into rectangles and triangles, find each area, then add (or subtract a piece that has been cut away).

▶ Split a compound shape into a rectangle and a triangle, find each area, then add.

Another type is a rectangle with a corner removed. A rectangular plot 14 m by 9 m has area $14\times 9=126$ m². If a triangular corner with legs 6 m and 4 m is cut away for a road, the road takes $\frac{1}{2}\times 6\times 4=12$ m², so the plot left is $126-12=114$ m².

Many questions also involve cost. First find the area, then multiply by the price per square metre. For the plot at Kasoa: area $=\frac{1}{2}\times 40\times 25=500$ m², and the clearing cost is $500\times 12=$ GH₵6,000.

Worked examples

Try each one yourself first, then open the solution and compare every step.