Area of a Triangle
Measurement · Area of a Triangle
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 10 Area of a Triangle
Lesson objectives
By the end of this topic, you should be able to:
- Find the area of a triangle given the base and perpendicular height.
- Solve problems involving area using the formula Area = ½ab sin C.
- Solve problems on the area of a triangle using Heron's formula.
- Select the appropriate formula for the information given.
Area of a Triangle
There are three formulas for the area of a triangle. Which one you use depends on what you are given.
a) Base and perpendicular height
The standard formula is A = ½ × base × height.
The height must be perpendicular to the base, not the slanting side. Using the slant gives an answer that is too large.
Any side may serve as the base, provided you use the height perpendicular to that side.
Use this formula when a height is given or can be found.
b) Two sides and the included angle
When two sides and the angle between them are known, use A = ½ ab sin C.
The angle C must lie between the sides a and b. An angle elsewhere in the triangle will not do.
This formula follows from the first. The perpendicular height is b sin C. So ½ ab sin C is ½ × base × height rewritten.
c) All three sides: Heron's formula
When all three sides are known but no angle is, use Heron's formula.
First find s, the semi-perimeter, which is half the sum of the sides.
Then the area is √(s(s−a)(s−b)(s−c)).
Work out s first and write down all four brackets before multiplying. Most errors here come from rushing that step.
d) Choosing the formula
| You are given | Use |
|---|---|
| Base and perpendicular height | ½ × base × height |
| Two sides and the angle between | ½ ab sin C |
| All three sides | Heron's formula |
Read the question and identify which case you are in before writing anything. Choosing wrongly wastes the most time in this topic.
e) Working backwards
Any of these can be rearranged to find a missing measurement.
Given the area and the base, the height is 2A ÷ base.
Given the area and two sides, sin C is 2A ÷ ab, and the angle follows.
f) Where this is used
Land measurement, where a plot's three sides can be paced but no height is available. Roofing and glazing. Any triangular region on a survey.
Words to know
- Perpendicular height -- the height measured at right angles to the chosen base.
- Included angle -- the angle lying between two named sides.
- Semi-perimeter -- half the sum of the three sides, written s.
- Heron's formula -- the area formula using only the three side lengths.
- Composite figure -- a shape made from two or more simpler shapes.
:::checkpoint Check yourself
- Find the area of a triangle with base 14 cm and perpendicular height 9 cm.
- Two sides are 6 cm and 9 cm with an included angle of 30°. Find the area.
- Find the area of a triangle with sides 3 cm, 4 cm and 5 cm using Heron's formula.
- Why must the angle in ½ ab sin C lie between the two sides? :::
Bridge to practice
The exercises begin with base and height, move through the included-angle formula and Heron's formula, and finish with selection, working backwards and composite figures. For every question, write down what is given before choosing any formula.