9.3PureCore

Areas of a Triangle

The familiar “½ × base × height” needs a height you often don’t have. This trig formula finds a triangle’s area straight from two sides and the angle between them — no perpendicular height required.

20 min Video by Zeeshan Zamurred Trigonometric Ratios

Edexcel AS Level Maths: 9.3 Areas of TrianglesWatch the full walkthrough before the notes below.

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What you'll be able to do

State the trig area formula
Find the area from two sides and the included angle
Rearrange to find a missing side or angle
Recognise the SAS condition the formula needs

The formula

For two sides $a$ and $b$ enclosing an angle $C$ , the area uses the sine of that included angle.

Area = \frac{1}{2} ab sin C

The angle

C

must be the one BETWEEN the two sides

a

and

b

1Area

= \frac{1}{2} (6) (8) sin 3 0^{\circ}

= 24 \times 0.5 = 12

Answer

12

square units

Tip — The angle must be enclosed by the two sides. If it is not, find it (or another pair) first.

Working backwards

Because the formula links area, two sides and an angle, you can rearrange it to find a missing side or the included angle when the area is given.

10 = \frac{1}{2} (10) (4) sin C = 20 sin C

sin C = 0.5

Answer

C = 3 0^{\circ}

(or

15 0^{\circ}

)

Formula recap

Area = \frac{1}{2} ab sin C

Area from two sides + included angle.

sin C = \frac{2 \times Area}{ab}

Rearranged for the angle.

Common mistakes to avoid

Using an angle that is not between the two sides.

The formula needs the INCLUDED angle (between sides a and b).

Using ½ × base × height when you don’t actually have the perpendicular height.

With two sides and the included angle, use ½ab·sin C instead.

Key takeaways

Area = ½ab·sin C, where C is the angle between sides a and b.
It needs the SAS arrangement (two sides + included angle).
Rearrange to find a missing side or the included angle from a known area.

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