Sine Rule and Cosine Rule
Area of Any Triangle

An important application of trigonometry is that of finding the area of a triangle with side lengths a and b and included angle θ.

The area (A) is given by absinθ°.

A = absinθ

Proof

If you construct the perpendicular from the vertex B to AC, then its length, p, is given by

p = asinθ

Thus the area of ABC is given by

area = × base × height
= × b × p
= × b × (asinθ)
= absinθ

as required.

Worked Examples

1

The diagram shows a circle of radius 64 cm.

The length of the chord AB is 100 cm.

(a)

Find the angle θ, to 2 d.p.

Show me

If AB = 100 cm then, by symmetry, BC = 50 cm.

sinθ =
θ = 51.38°
(b)

Find the area of triangle OAB.

Show me

The area of the triangle OAB is × 642 × sin2θ = 1997 cm2
2

Given that sinθ = , 0° ≤ θ ≤ 90°

(a)

Express in fractional or surd form the value of cosθ.

Show me

sin2θ + cos2θ = 1

so

cos2θ = 1 − ⎛
⎝⎞
⎠2
= 1 −
=

Hence

cosθ =

[ An alternative approach is to consider the right angled triangle, as shown.

If the lengths

BC = , AC = 2

then clearly sinθ = . We can find the length AB:

AB2 = AC2 − CB2 (Pythagoras)
= 4 − 3 = 1

So AB = 1 and cosθ = = ]
(b)

Show that the area of triangle CDE is 150 square units, where CD = 30 units and DE = 20 units.

Show me

In the given triangle,

Area = × CD × DEsinθ
= × 30 × 20 ×
= 150
(c)

Calculate the length of the side EC.

Show me

We can find the length EC using the cosine rule, namely

EC2 = DE2 + DC2 − 2 × DE × DC × cosθ
= 202 + 302 − 2 × 20 × 30 ×
= 400 + 900 − 600
= 700
EC ≈ 26.5

Exercises

Find the area of the shaded region in each of the following figures.

Round your answers to 2 d.p.

(a)
cm2
(b)
cm2

Find the area of the shaded region.

Round your answers to 2 d.p.

(a)
cm2
(b)
cm2

Find the area of the shaded region.

Round to 1 d.p.

(a)
cm2
(b)
cm2

In the diagram above, not drawn to scale, ST = 5 cm, TW = 9 cm and STW = 52°.

Calculate

(a)

the length of SW

cm (to 2 d.p.)
(b)

the area of Δ STW.

cm2 (to 2 d.p.)

In the diagram above, not drawn to scale, P and Q are midpoints of the sides XY and XZ of triangle XYZ. Given that XP = 7.5 cm, XQ = 4.5 cm and the area of triangle XPQ = 13.5 cm2, calculate

Round your answers to the nearest whole number.

(a)

the size of angle PXQ, expressing your answer correct to the nearest degree.

°
(b)

the area of triangle YXZ.

cm2

On the diagram below, not drawn to scale, TU = 8 m, TW = 10 m, VW = 11 m, angle UTW = 60° and angle WUV = 40°.

Calculate

(a)
the length of UW m (to 2 d.p.)
(b)
the size of the angle UVW ° (to 1 d.p.)
(c)
the area of triangle TUW. m2 (to 2 d.p.)