Year 11 General Applications Of Trigonometry

The Area Of A Triangle

Q: What is the formula for the area of a triangle with two sides and an angle?

Area equals one half a b sin C, where a and b are the two known sides and C is the angle between them. The angle must be the included angle, not any other angle of the triangle.

Q: What is the included angle?

The included angle is the angle between two specified sides. For example, if a triangle has sides of 7 cm and 9 cm meeting at one vertex, the included angle is the angle at that vertex. It's not one of the other two angles of the triangle.

Q: What is the semi-perimeter?

The semi-perimeter s is half the perimeter: s equals (a plus b plus c) divided by 2. It appears in Heron's formula. For a triangle with sides 7, 8, 9, the perimeter is 24 and the semi-perimeter is 12.

Q: How do I find the area of an equilateral triangle?

All angles in an equilateral triangle are 60 degrees, so use area equals one half a squared sin 60 degrees where a is the side length. Equivalently, the exact area is root 3 over 4 times a squared. For a equals 10 cm, area is approximately 43.3 square centimetres.

Q: How do I find an angle of a triangle from its area?

Use area equals one half a b sin C and solve for sin C: rearrange to sin C equals 2 times area divided by ab, then apply sin inverse. Two answers are possible (one acute, one obtuse) — the question usually says which one to choose.

20 practice questions 2 video lessons Theory + worked examples

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Area of a Triangle, Given 3 Sides, Heron's Formula Watch
Area of a Triangle - 1/2 absin(C) - GCSE Higher Maths Watch

Theory

When two sides of a triangle and the angle between them are known, the area is $A = \frac{1}{2} a b \sin C$ . When all three sides are known, use Heron's formula: $A = \sqrt{s (s - a) (s - b) (s - c)}$ , where $s$ is the semi-perimeter $\frac{a + b + c}{2}$ .

The familiar formula $A = \frac{1}{2} \cdot base \cdot height$ only works when you know the perpendicular height. For triangles where you know two sides and an angle, or all three sides, there are two more powerful formulas:

Two sides and the included angle — use $A = \frac{1}{2} a b \sin C$ , where $a$ and $b$ are the two sides and $C$ is the angle between them.
All three sides known — use Heron's formula. First compute the semi-perimeter $s = \frac{a + b + c}{2}$ , then $A = \sqrt{s (s - a) (s - b) (s - c)}$ .

The included angle is the angle between two sides — not any random angle of the triangle. If a question gives you a non-included angle, you have to find an unknown side first (using the sine rule) before you can apply the area formula.

Two sides

a, b

with included angle

C

A = \frac{1}{2} a b \sin C

All three sides known:

A = \sqrt{s (s - a) (s - b) (s - c)}

Area from two sides and the included angle

A = \frac{1}{2} a b \sin C

A = \frac{1}{2} a b \sin C

Heron's formula (three sides known)

First compute the semi-perimeter $s$ :

s = \frac{a + b + c}{2}

s = \frac{a + b + c}{2}

Then the area is:

A = \sqrt{s (s - a) (s - b) (s - c)}

A = \sqrt{s (s - a) (s - b) (s - c)}

Choosing the right formula

What you know	Use
Base and perpendicular height	$A = \frac{1}{2} b h$
Two sides and the included angle	$A = \frac{1}{2} a b \sin C$
All three sides	Heron's: $A = \sqrt{s (s - a) (s - b) (s - c)}$

          Quadrilateral areas: split the quadrilateral into two triangles by drawing a diagonal, find each triangle's area separately, then add them.
        

How to find the area of any triangle

Identify what you're given: base + height, two sides + included angle, or three sides.
Choose the matching formula from the table above.
Substitute the values — for Heron's, compute $s$ first, then $s - a$ , $s - b$ , $s - c$ .
Evaluate with the calculator in degree mode, and write the answer with the correct units (e.g.\ cm $^{2}$ , m $^{2}$ ).

Example 1 — Two sides + angle

A triangle has sides

12

cm and

9

cm with an included angle of

58^{\circ}

. Find its area (1 dp).

Solution

Sketch the triangle with the angle between the two known sides:

$A$	$=$	$\frac{1}{2} a b \sin C$
$A$	$=$	$\frac{1}{2} \times 12 \times 9 \times \sin 58^{\circ}$
$A$	$\approx$	$45.8 {cm}^{2}$

A = \frac{1}{2} \times 12 \times 9 \times \sin 58 ° \approx 45.8 cm ²

Example 2 — Equilateral triangle

Find the area of an equilateral triangle with side

10

cm (1 dp).

Solution

All angles in an equilateral triangle are $60^{\circ}$ . Use any two sides:

$A$	$=$	$\frac{1}{2} \times 10 \times 10 \times \sin 60^{\circ}$
$A$	$=$	$50 \sin 60^{\circ}$
$A$	$\approx$	$43.3 {cm}^{2}$

A = 50 \sin 60 ° \approx 43.3 cm ²

Example 3 — Heron's formula

Find the area of a triangle with sides

7

8

9

(1 dp).

Solution

Compute the semi-perimeter, then apply Heron's:

$s$	$=$	$\frac{7 + 8 + 9}{2} = 12$
$A$	$=$	$\sqrt{12 \cdot 5 \cdot 4 \cdot 3}$
$A$	$=$	$\sqrt{720}$
$A$	$\approx$	$26.8 u^{2}$

A = \sqrt{720} \approx 26.8

Example 4 — Find an angle from area

A triangle has sides

8

cm and

11

cm with an area of

30

^{2}

. Find the included angle (acute, nearest degree).

Solution

Substitute and solve for $\sin C$ , then apply $\sin^{- 1}$ :

$\frac{1}{2} \times 8 \times 11 \times \sin C$	$=$	$30$
$44 \sin C$	$=$	$30$
$\sin C$	$=$	$\frac{30}{44} \approx 0.6818$
$C$	$=$	$\sin^{- 1} (0.6818)$
$C$	$\approx$	$43^{\circ}$

C = \sin^{- 1} (0.6818) \approx 43 °

Common pitfalls

The angle must be the included angle. For

A = \frac{1}{2} a b \sin C

C

must be the angle between sides

a

and

b

. If the angle isn't included, use the sine rule first to find a missing side.

Heron's: write each $s - side$ carefully. A negative result means you've subtracted the wrong way.

s

is always at least as large as the longest side, so

s - a

s - b

s - c

are all positive.

Two angles can give the same area. If

\sin C = 0.5

, then

C = 30^{\circ}

150^{\circ}

— both produce the same area. The question may specify "acute" or "obtuse" to disambiguate.

Units squared. Area is measured in square units: cm

^{2}

, m

^{2}

, km

^{2}

. Don't drop the squared.

Frequently asked questions

What is the formula for the area of a triangle with two sides and an angle?

$A = \frac{1}{2} a b \sin C$ , where $a$ and $b$ are the two known sides and $C$ is the angle between them. The angle must be the included angle, not any other angle of the triangle.

What is Heron's formula?

Heron's formula finds the area of a triangle from its three side lengths alone, with no angle needed. First find the semi-perimeter $s = \frac{a + b + c}{2}$ , then $A = \sqrt{s (s - a) (s - b) (s - c)}$ .

What is the included angle?

The included angle is the angle between two specified sides. For example, if a triangle has sides of $7$ cm and $9$ cm meeting at one vertex, the included angle is the angle at that vertex. It's NOT one of the other two angles of the triangle.

What is the semi-perimeter?

The semi-perimeter $s$ is half the perimeter: $s = \frac{a + b + c}{2}$ . It appears in Heron's formula. For a triangle with sides $7$ , $8$ , $9$ , the perimeter is $24$ and the semi-perimeter is $12$ .

How do I find the area of an equilateral triangle?

All angles in an equilateral triangle are $60^{\circ}$ , so use $A = \frac{1}{2} a^{2} \sin 60^{\circ}$ where $a$ is the side length. Equivalently, the exact area is $A = \frac{\sqrt{3}}{4} a^{2}$ . For $a = 10$ cm, $A \approx 43.3$ cm $^{2}$ .

How do I find an angle of a triangle from its area?

Use $A = \frac{1}{2} a b \sin C$ and solve for $\sin C$ : rearrange to $\sin C = \frac{2 A}{a b}$ , then apply $\sin^{- 1}$ . Two answers are possible (one acute, one obtuse) — the question usually says which one to choose.

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The Sine Rule

The Area Of A Triangle

📖 Theory

Area from two sides and the included angle

Heron's formula (three sides known)

Choosing the right formula

How to find the area of any triangle

Common pitfalls

Frequently asked questions

Theory