GCSE Maths Pythagoras & trig questions by topic

Finding hypotenuses.

1. (a)A right-angled triangle has legs of 8 cm and 15 cm. Find the hypotenuse.[2]
2. (b)A right-angled triangle has legs of 7 cm and 24 cm. Find the hypotenuse.[2]

Finding shorter sides.

1. (a)A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.[2]
2. (b)A right-angled triangle has hypotenuse 10 cm and one leg 6 cm. Find the other leg, to 1 decimal place.[2]

Trigonometry for sides.

1. (a)A right-angled triangle has an angle of 32° and a hypotenuse of 15 cm. Find the side opposite the 32° angle, to 2 decimal places.[2]
2. (b)A right-angled triangle has an angle of 42° with an adjacent side of 9 cm. Find the hypotenuse, to 2 decimal places.[2]

Trigonometry for angles.

1. (a)In a right-angled triangle the side opposite an angle is 6 cm and the hypotenuse is 14 cm. Find the angle, to 1 decimal place.[2]
2. (b)In a right-angled triangle the opposite side is 5 cm and the adjacent side is 8 cm. Find the angle, to 1 decimal place.[2]

A patio and its diagonal slab.

1. (a)A rectangular patio measures 6 m by 8 m. A straight slab runs diagonally across it. Work out the length of the diagonal.[3]
2. (b)Work out the perimeter of the patio.[1]

A beam braces a wall.

1. (a)A 13 m beam reaches from the ground to a wall, standing 5 m out from the wall’s base. Work out how high up the wall the beam reaches.[2]
2. (b)Calculate, to 1 decimal place, the angle at the foot of the beam.[2]

A ladder against a wall.

1. (a)A 4.5 m ladder leans against a vertical wall at 68° to the horizontal ground. How high up the wall does it reach? Give your answer to 2 decimal places.[3]
2. (b)How far is the foot of the ladder from the wall, to 2 decimal places?[2]

Pythagoras then trigonometry.

1. (a)A right-angled triangle has legs 9 cm and 12 cm. Find the hypotenuse.[2]
2. (b)Find the angle between the hypotenuse and the 12 cm side, to 1 decimal place.[2]
3. (c)Explain why Pythagoras must be used before part (b).[1]

Exact values in surd arithmetic. (Higher)

1. (a)Write down the exact value of sin 30°.[1]
2. (b)Write down the exact value of cos 60°.[1]
3. (c)Write down the exact value of tan 45°.[1]
4. (d)Evaluate sin 45° × √2, leaving your answer in exact form.[2]

A ship’s journey.

1. (a)A ship sails 15 km due east, then 8 km due north. How far is it from its starting point?[2]
2. (b)Find the bearing of the ship from its starting point, to 1 decimal place.[3]

A ramp’s angle and surface.

1. (a)A ramp rises 1.2 m over a horizontal distance of 5 m. Work out the angle it makes with the ground, to 1 decimal place.[2]
2. (b)Work out the length of the ramp surface, to 2 decimal places.[3]

Sine rule. (Higher)

1. (a)In triangle ABC, angle A = 38°, angle B = 63° and a = 9 cm. Find angle C.[1]
2. (b)Find side b, to 1 decimal place.[3]
3. (c)Explain why the sine rule can be used for this triangle.[2]

Cosine rule and area. (Higher)

1. (a)A triangle has sides 7 cm and 10 cm enclosing an angle of 49°. Find the third side, to 2 decimal places.[4]
2. (b)Find the area of the triangle, to 1 decimal place.[2]

A cuboid’s diagonals. (Higher)

1. (a)A cuboid measures 5 cm by 6 cm by 12 cm. Find the diagonal of the 5 cm × 6 cm face, to 2 decimal places.[2]
2. (b)Find the space diagonal (corner to opposite corner), to 2 decimal places.[3]
3. (c)Explain why Pythagoras’ theorem is used twice.[1]

The ambiguous case. (Higher)

1. (a)In triangle PQR, p = 11 cm, q = 14 cm and angle P = 42°. Use the sine rule to find angle Q, to 1 decimal place.[4]
2. (b)State the other possible value of angle Q and explain why it is also valid.[2]

A surd sum and a Pythagorean identity. (Higher)

1. (a)Write sin 30° + sin 60° as a single fraction in surd form.[3]
2. (b)Explain, using a right-angled triangle, why sin²x + cos²x = 1 for every angle x.[3]

An oblique triangle is solved by the sine rule. (Higher)

1. (a)In triangle ABC, angle A = 40°, angle C = 65° and AB = 9 cm. Work out angle B.[1]
2. (b)Use the sine rule to set up an equation for side BC.[2]
3. (c)Work out BC to 2 decimal places.[3]

A cuboid hides an exact diagonal. (Higher)

1. (a)A cuboid measures 2 cm by 3 cm by 6 cm. Work out the diagonal of the 2 cm by 3 cm face, to 2 decimal places.[2]
2. (b)Use your answer to work out the space diagonal of the cuboid.[3]
3. (c)Explain why the space diagonal comes out exact.[1]