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.

11 practice questions 2 video lessons Theory + worked examples

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.