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Maths · More trigonometry

Calculating areas and the sine rule

Use the labelling of triangles, area \(= \tfrac{1}{2}ab\sin C\), and the sine rule to find missing sides and angles in non-right-angled triangles.

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  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the area of a triangle?

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    \(\tfrac{1}{2} \times \text{base} \times \text{height}\)

  2. 2

    What do the angles in a triangle add up to?

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    \(180^\circ\)

  3. 3

    What is \(\sin^{-1}(0.5)\)?

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    \(30^\circ\)

  4. 4

    Which side is opposite the right angle?

    Show answerHide answer

    The hypotenuse

  5. 5

    What does SOH CAH TOA help with?

    Show answerHide answer

    Right-angled triangles only

Learning Objectives

  1. 1Label a triangle with lower-case sides opposite capital angles.
  2. 2Find the area of any triangle using \(\tfrac{1}{2}ab\sin C\).
  3. 3Use the sine rule to find a missing side.
  4. 4Use the sine rule to find a missing angle.

SINE RULE

In any triangle, the sides and the sines of their opposite angles are in proportion.

\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\). Flip it to find an angle: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\).

Which Formula?

  • Area

    \(\tfrac{1}{2}ab\sin C\): two sides and the angle between them.

  • Sine rule (side)

    Two angles and a side, need another side.

  • Sine rule (angle)

    Two sides and a non-included angle, need an angle.

  • Neither

    Three sides, or two sides and the included angle: use the cosine rule (next lesson).

Area of a Triangle

Find the area of a triangle with sides \(a = 8\) cm and \(b = 11\) cm and angle \(C = 50^\circ\) between them. Give your answer to 3 significant figures.

Show the solutionHide the solution
  1. 1 Write the formula Area \(= \tfrac{1}{2}ab\sin C\)
  2. 2 Substitute \(\tfrac{1}{2} \times 8 \times 11 \times \sin 50^\circ\)
  3. 3 Work out \(44 \times 0.7660 = 33.7\)

AnswerArea \(= 33.7\) cm\(^2\)

Sine Rule: Finding a Side

In triangle ABC, \(A = 40^\circ\), \(B = 65^\circ\) and \(a = 8\) cm. Find \(b\).

Show the solutionHide the solution
  1. 1 Set up \(\dfrac{b}{\sin 65^\circ} = \dfrac{8}{\sin 40^\circ}\)
  2. 2 Rearrange \(b = \dfrac{8 \times \sin 65^\circ}{\sin 40^\circ}\)
  3. 3 Work out \(b = 11.28\)

Answer\(b = 11.3\) cm (3 s.f.)

Sine Rule: Finding an Angle

In triangle ABC, \(a = 7\) cm, \(b = 9\) cm and angle \(A = 35^\circ\). Find angle \(B\).

Show the solutionHide the solution
  1. 1 Set up \(\dfrac{\sin B}{9} = \dfrac{\sin 35^\circ}{7}\)
  2. 2 Rearrange \(\sin B = \dfrac{9 \times \sin 35^\circ}{7} = 0.7375\)
  3. 3 Inverse sine \(B = \sin^{-1}(0.7375) = 47.5^\circ\)

Answer\(B = 47.5^\circ\) (1 d.p.)

Setting Out

Put the unknown on top.

  • Missing side

    Write the rule with sides on top: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\).

  • Missing angle

    Write the rule with sines on top: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\).

  • Diagram

    Mark which side is opposite which angle before you start.

  • Check

    The biggest angle is opposite the longest side.

Choose the Method

For each, say whether you would use the area formula, the sine rule, or neither yet, and why. (a) Two angles and one side, find another side. (b) Two sides and the angle between them, find the area. (c) Three sides, find an angle. (d) Two sides and an angle not between them, find another angle.

1. List what is given.

2. Choose the formula.

A good answer shows: (a) Sine rule. (b) Area \(= \tfrac{1}{2}ab\sin C\). (c) Neither yet: cosine rule. (d) Sine rule.

Can I...?

  1. 1Label sides and angles.
  2. 2Write the area formula.
  3. 3Find the area of a triangle.
  4. 4Write the sine rule.
  5. 5Find a missing side.
  6. 6Find a missing angle.
  7. 7Put the unknown on top.
  8. 8Check the answer is sensible.

Summary & Exam Focus

  • Area \(= \tfrac{1}{2}ab\sin C\).
  • Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\).
  • Use the sine rule with two angles and a side, or two sides and a non-included angle.
  • Check side lengths against angles.

Exam focus

In triangle ABC, \(AB = 7\) cm, angle \(B = 40^\circ\) and angle \(C = 75^\circ\). Work out the length of \(AC\). Give your answer correct to 3 significant figures. (3 marks) (3 marks)

Write the sine rule with the unknown side on top. Keep your calculator in degrees.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Sine rule
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).
Opposite
Across the triangle from an angle, not touching it.
Included angle
The angle between two given sides.
Scalene
A triangle with no equal sides.
Vertex
A corner of a shape.
Non-right-angled
A triangle with no \(90^\circ\) angle.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Work out 3 marks

    In triangle ABC, \(AB = 7\) cm, angle \(ABC = 40^\circ\) and angle \(ACB = 75^\circ\). Work out the length of \(AC\). Give your answer correct to 3 significant figures.

    Triangle ABC with AB equal to 7 cm, angle B equal to 40 degrees and angle C equal to 75 degrees, with side AC to be found.
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    Model answer

    \(AC = \dfrac{7 \sin 40^\circ}{\sin 75^\circ} = 4.66\) cm

    Mark scheme

    • \(\dfrac{AC}{\sin 40^\circ} = \dfrac{7}{\sin 75^\circ}\) — M1
    • \(\dfrac{7 \times \sin 40^\circ}{\sin 75^\circ}\) — M1
    • 4.66 — A1
  2. Question 2 Work out 3 marks

    In triangle ABC, angle \(A = 52^\circ\), angle \(B = 71^\circ\) and \(BC = 9.4\) cm. Work out the length of \(AC\). Give your answer correct to 3 significant figures.

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    Model answer

    \(AC = \dfrac{9.4 \sin 71^\circ}{\sin 52^\circ} = 11.3\) cm

    Mark scheme

    • \(\dfrac{AC}{\sin 71^\circ} = \dfrac{9.4}{\sin 52^\circ}\) — M1
    • \(\dfrac{9.4 \times \sin 71^\circ}{\sin 52^\circ}\) — M1
    • 11.3 — A1
  3. Question 3 Work out 3 marks

    In triangle ABC, \(BC = 7\) cm, \(AC = 9\) cm and angle \(BAC = 35^\circ\). Work out the size of angle \(ABC\). Give your answer correct to 1 decimal place. (Angle \(ABC\) is acute.)

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    Model answer

    \(\sin B = \dfrac{9 \sin 35^\circ}{7} = 0.7375\), so \(B = 47.5^\circ\)

    Mark scheme

    • \(\dfrac{\sin B}{9} = \dfrac{\sin 35^\circ}{7}\) — M1
    • \(\sin B = 0.7375...\) — M1
    • 47.5 — A1
  4. Question 4 Work out 2 marks

    A triangle has two sides of length 9.5 cm and 6.2 cm with an angle of \(112^\circ\) between them. Work out the area of the triangle. Give your answer correct to 3 significant figures.

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    Model answer

    \(\tfrac{1}{2} \times 9.5 \times 6.2 \times \sin 112^\circ = 27.3\) cm\(^2\)

    Mark scheme

    • \(\tfrac{1}{2} \times 9.5 \times 6.2 \times \sin 112^\circ\) — M1
    • 27.3 — A1
  5. Question 5 Work out 4 marks

    The area of a triangle is 25 cm\(^2\). Two of its sides are 5 cm and 10 cm and the angle between them is acute. Work out the size of the angle between these two sides. Give your answer correct to 1 decimal place.

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    Model answer

    \(\tfrac{1}{2} \times 5 \times 10 \times \sin C = 25\), so \(\sin C = 1\), giving \(C = 90^\circ\).

    Mark scheme

    • \(\tfrac{1}{2} \times 5 \times 10 \times \sin C = 25\) — M1
    • \(25 \sin C = 25\) — M1
    • \(\sin C = 1\) — A1
    • 90.0 — A1
  6. Question 6 Explain 3 marks

    In a triangle, angle \(A = 30^\circ\), \(a = 6\) cm and \(b = 10\) cm. Show that \(\sin B = \dfrac{5}{6}\).

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    Model answer

    \(\dfrac{\sin B}{10} = \dfrac{\sin 30^\circ}{6}\), so \(\sin B = \dfrac{10 \times 0.5}{6} = \dfrac{5}{6}\).

    Mark scheme

    • Sine rule set up — M1
    • \(\sin 30^\circ = 0.5\) — M1
    • Completes the proof — A1

Quick check

  1. In triangle ABC, side \(a\) is opposite...

    1. AAngle A
    2. BAngle B
    3. CAngle C
    4. DThe right angle
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    A: Angle A

    Lower-case a is the side opposite capital A.

  2. The area of a triangle with sides \(a\), \(b\) and included angle \(C\) is...

    1. A\(ab\sin C\)
    2. B\(\tfrac{1}{2}ab\cos C\)
    3. C\(\tfrac{1}{2}ab\sin C\)
    4. D\(\tfrac{1}{2}a\sin C\)
    Show answerHide answer

    C: \(\tfrac{1}{2}ab\sin C\)

    \(\tfrac{1}{2}ab\sin C\).

  3. Which lets you find a side using the sine rule?

    1. AThree sides
    2. BTwo angles and a side
    3. CTwo sides and the angle between them
    4. DThree angles
    Show answerHide answer

    B: Two angles and a side

    Two angles and a side.

  4. To find an angle with the sine rule, write it as...

    1. A\(\dfrac{a}{\sin A}\) with a on top
    2. B\(a \sin A\)
    3. C\(\dfrac{a}{b}\) only
    4. D\(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\)
    Show answerHide answer

    D: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\)

    Sines on top: \(\dfrac{\sin A}{a} = \dfrac{\sin B}{b}\).

  5. A triangle has sides 4 and 6 with an angle of \(30^\circ\) between them. Its area is...

    1. A6
    2. B12
    3. C3
    4. D24
    Show answerHide answer

    A: 6

    \(\tfrac{1}{2} \times 4 \times 6 \times 0.5 = 6\).

  6. In a triangle, the longest side is opposite...

    1. AThe smallest angle
    2. BThe right angle
    3. CThe largest angle
    4. DAny angle
    Show answerHide answer

    C: The largest angle

    The largest angle.

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