GCSE Maths · Geometry and measures

Cosine Rule GCSE Questions, Worked Examples and Answers

The cosine rule connects all three sides of any triangle with one angle. Use it to find a side from two sides and their included angle, or to find an angle when all three sides are known.

Edexcel · AQA · OCRHigher tierOriginal diagrams · 10 questions

Ask Ari about the cosine rule Start the questions

Free AI tutor · GCSE the cosine rule

Practise GCSE the cosine rule for free with an AI tutor

Ask Ari for an explanation or work through an original exam-style question together.

Hi, I’m Ari. I can label a cosine-rule triangle, help you decide whether the rule fits or check a side or angle calculation.

Ari is an AI tutor and can make mistakes. Use the worked answers below to check important results.

Match each angle with its opposite side

What you need to know about the cosine rule

Imagine two paths leaving the same point. Knowing both path lengths is not enough to find the gap between their endpoints: the angle between the paths matters too. The cosine rule finds this third side, even when there is no right angle. We label the angle between the paths A, their lengths b and c, and the side facing A with the matching lower-case letter a.

Two sides + included angle

Read the triangle before the formula

To find side a, use angle A between sides b and c. Cos A is the cosine of that angle, found with the cosine key in degree mode. The rule adjusts b² + c² for how far the two sides open apart; at 90°, cos A = 0 and the adjustment vanishes, leaving Pythagoras.

Targetside aopposite angle A

Knownsides b, c and angle Athe included angle sits between b and c

Relationshipa² = b² + c² − 2bc cos Athen take the positive square root

Lower-case side a is opposite upper-case angle A. This matching pair determines which term appears alone in the cosine rule.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Find a missing side

What the problem asks: Two side lengths and the angle between them are known.

How to solve it: Match the missing side with its opposite angle, substitute into a² = b² + c² − 2bc cos A, then take the positive square root.

Find a missing angle

What the problem asks: All three side lengths are known.

How to solve it: Rearrange to cos A = (b² + c² − a²) ÷ (2bc), then use inverse cosine in degree mode. The cos⁻¹ key asks ‘which angle has this cosine?’; it does not mean 1 ÷ cos.

Solve a contextual distance

What the problem asks: Two journeys, bearings or forces form two known sides with an included angle.

How to solve it: Draw the triangle, calculate the included angle carefully, then use the missing-side form.

Complete a multi-step shape

What the problem asks: A diagonal divides a compound shape into triangles.

How to solve it: Use the cosine rule in the triangle with sufficient information, then carry the result into the next valid relationship.

A reliable routine

Method for a cosine-rule calculation

Use the rule to find a side when two sides and the angle between them are known, or to find an angle when all three sides are known. These arrangements are often shortened to SAS and SSS. The rule works for any triangle and extends Pythagoras by accounting for the included angle.

Sketch or mark the target side or angle.
Label the target pair with matching lower-case and capital letters.
Choose the side form or rearranged angle form.
Substitute with brackets and check the calculator is in degree mode.
Keep the full calculator value until the final requested accuracy.

Check: If A = 90°, cos A = 0, so the final term disappears and the cosine rule becomes Pythagoras’ theorem.

Fully worked

Cosine rule GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Find a side from SAS

Higher only3 marks

Question

In triangle $ABC$ , $b=8$ cm, $c=11$ cm and $A=60^\circ$ . Find $a$ to 3 significant figures.

Lower-case side a is opposite upper-case angle A. This matching pair determines which term appears alone in the cosine rule.

Side $a$ is opposite the known angle.

a^2=8^2+11^2-2(8)(11)\cos60^\circ

a^2=64+121-88

a^2=185-88

a^2=97

a=\sqrt{97}

\boxed{a=9.85\text{ cm}\quad(3\text{ s.f.})}

Example 2

Find an angle from SSS

Higher only4 marks

Question

A triangle has $a=10$ cm, $b=7$ cm and $c=12$ cm. Find angle $A$ to 1 decimal place.

Rearrange the rule for the angle opposite side $a$ .

\cos A=\frac{b^2+c^2-a^2}{2bc}

\cos A=\frac{7^2+12^2-10^2}{2(7)(12)}

\cos A=\frac{93}{168}

A=\cos^{-1}\left(\frac{93}{168}\right)

\boxed{A=56.4^\circ}

Example 3

Use an obtuse included angle

Higher only3 marks

Question

Two sides are $6$ cm and $9$ cm with included angle $120^\circ$ . Find the opposite side to 3 significant figures.

Let the opposite side be $a$ .

a^2=6^2+9^2-2(6)(9)\cos120^\circ

Because $\cos120^\circ=-0.5$ ,

a^2=36+81+54=171

a=\sqrt{171}

\boxed{a=13.1\text{ cm}\quad(3\text{ s.f.})}

Example 4

Find a distance between two routes

Higher only4 marks

Question

Two straight routes leave the same point. One is $7$ km and the other is $9$ km. The angle between them is $70^\circ$ . Find the distance between their endpoints to 3 significant figures.

The required distance is opposite the included angle.

d^2=7^2+9^2-2(7)(9)\cos70^\circ

d^2\approx86.9055

d\approx9.32231

\boxed{9.32\text{ km}\quad(3\text{ s.f.})}

Example 5

Find a triangle perimeter

Higher only4 marks

Question

Two sides of a triangle are $5$ cm and $8$ cm and their included angle is $45^\circ$ . Find the perimeter to 3 significant figures.

First find the third side $a$ .

a^2=5^2+8^2-2(5)(8)\cos45^\circ

a=5.694862\ldots

Now add all three sides.

5+8+5.694862\ldots=18.694862\ldots

\boxed{18.7\text{ cm}\quad(3\text{ s.f.})}

10 original questions · total 36 marks

Cosine rule GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 45 minutes · Label the opposite side–angle pair, show the substituted cosine rule and keep calculator values unrounded until the final answer. · answers start collapsed

Find a missing side

Higher only3 marks

Sides $5$ cm and $7$ cm include an angle of $60^\circ$ . Find the opposite side to 3 significant figures.

Show worked answer

a^2=5^2+7^2-2(5)(7)\cos60^\circ

a^2=39

a=\sqrt{39}

\boxed{6.24\text{ cm}}

Connect to Pythagoras

Higher only3 marks

Sides $9$ cm and $12$ cm include an angle of $90^\circ$ . Use the cosine rule to find the opposite side.

Show worked answer

a^2=9^2+12^2-2(9)(12)\cos90^\circ

Since $\cos90^\circ=0$ ,

a^2=81+144=225

\boxed{a=15\text{ cm}}

Find an angle

Higher only4 marks

A triangle has sides $a=13$ , $b=8$ and $c=11$ . Find angle $A$ to 1 decimal place.

Show worked answer

\cos A=\frac{8^2+11^2-13^2}{2(8)(11)}

\cos A=\frac{16}{176}

A=\cos^{-1}\left(\frac{16}{176}\right)

\boxed{A=84.8^\circ}

Find an obtuse angle

Higher only4 marks

A triangle has sides $5$ cm, $6$ cm and $9$ cm. Find the angle opposite the $9$ cm side to 1 decimal place.

Show worked answer

\cos A=\frac{5^2+6^2-9^2}{2(5)(6)}

\cos A=-\frac13

A=\cos^{-1}\left(-\frac13\right)

\boxed{A=109.5^\circ}

Use an isosceles triangle

Higher only3 marks

Two equal sides are each $10$ cm and their included angle is $40^\circ$ . Find the base to 3 significant figures.

Show worked answer

a^2=10^2+10^2-2(10)(10)\cos40^\circ

a\approx6.84040

\boxed{6.84\text{ cm}}

Find a separation

Higher only4 marks

Two boats leave a harbour on routes of $12$ km and $15$ km with an angle of $130^\circ$ between them. Find their separation to 3 significant figures.

Show worked answer

d^2=12^2+15^2-2(12)(15)\cos130^\circ

d^2\approx600.403

d\approx24.5031

\boxed{24.5\text{ km}}

Find a perimeter

Higher only4 marks

Two sides are $9$ cm and $13$ cm with included angle $50^\circ$ . Find the triangle perimeter to 3 significant figures.

Show worked answer

a^2=9^2+13^2-2(9)(13)\cos50^\circ

a\approx9.97936

P=9+13+9.97936

\boxed{P=32.0\text{ cm}\quad(3\text{ s.f.})}

Choose the correct opposite pair

Higher only3 marks

In triangle $PQR$ , $PQ=8$ , $PR=10$ and angle $QPR=35^\circ$ . Write and evaluate an expression for $QR^2$ .

Show worked answer

$QR$ is opposite the included $35^\circ$ angle.

QR^2=8^2+10^2-2(8)(10)\cos35^\circ

\boxed{QR^2\approx32.936}

Explain the right-angle case

Higher only3 marks

Explain why the cosine rule becomes Pythagoras’ theorem when $A=90^\circ$ .

Show worked answer

a^2=b^2+c^2-2bc\cos90^\circ

Since $\cos90^\circ=0$ , the final term is zero.

a^2=b^2+c^2

This is $\boxed{\text{Pythagoras' theorem}}$ .

Form and solve a quadratic

Higher only5 marks

A triangle has sides $x$ , $7$ and $10$ . The angle opposite the $10$ side is $60^\circ$ . Find the positive value of $x$ to 3 significant figures.

Show worked answer

Use $10$ as the side opposite $60^\circ$ .

10^2=x^2+7^2-2(x)(7)\cos60^\circ

100=x^2+49-7x

x^2-7x-51=0

The positive quadratic-formula solution is

x=\frac{7+\sqrt{253}}2

\boxed{x=11.5\quad(3\text{ s.f.})}

Examiner-style feedback

Common the cosine rule mistakes

Pairing a with the wrong angle

Side a must be opposite angle A. Relabel the triangle before choosing the formula line.

Using a non-included angle for SAS

For a missing side, the known angle must sit between the two known sides.

Forgetting inverse cosine

The rearranged expression gives cos A, not A. Apply cos⁻¹ in degree mode.

Rounding the new side too soon

A rounded diagonal can make every later step inaccurate. Store or retain the full calculator value.

30-SECOND RECAP

Choose, carry out, check

Translate the wording before you reach for a rule.

Match side a with opposite angle A.
SAS finds a side; SSS finds an angle.
Use degree mode.
Round only the final result.

Quick answers

Cosine rule FAQ

When do I use the cosine rule?

Use it for two sides and their included angle when finding a side, or for all three sides when finding an angle.

What is the included angle?

It is the angle between the two known sides.

Can the cosine rule find an obtuse angle?

Yes. A negative cosine value produces an angle greater than 90° when inverse cosine is used.

How is it related to Pythagoras?

When the included angle is 90°, cos 90° = 0 and the cosine rule reduces to a² = b² + c².

Build connected skills

What to revise next

Right-angle prerequisite

Pythagoras' theorem

Understand the right-triangle case before using the cosine rule.

Revise pythagoras' theorem

Opposite pairs

Sine rule

Find a side or angle when a complete opposite pair is known.

Revise sine rule

Right triangles

Trigonometry

Use sine, cosine and tangent in right-angled triangles.

Revise trigonometry

Known angle-side pairs

Exact trig values

Use exact values for special angles without rounding.

Revise exact trig values

Contextual direction

Bearings

Build the included angle from three-figure bearings.

Revise bearings

Content standards

Curriculum and rights review

Reviewed 5 September 2026 against DfE content G22 and current Pearson Edexcel, AQA and OCR GCSE Mathematics specifications. All questions and diagrams are original.

Official specification references