Grade Ladder: Geometry and Measures
18 questions climbing from grade 1 to grade 9 across the geometry content of the Edexcel GCSE, two at each grade. Each question shows its grade, its specification code and whether it is non-calculator.
How to use this ladderFoundation students: work through Grades 1–5. Higher students: start at Grade 4 and push on to 9. Questions marked Non-calculator should be done without a calculator, as in Paper 1.
Grades 1–3 · Foundation
- (1 mark)Grade 1G3Non-calculatorTwo angles lie on a straight line. One is 65°. Find the other.
- (1 mark)Grade 1G17Non-calculatorFind the perimeter of a rectangle 7 cm by 4 cm.
- (1 mark)Grade 2G16Non-calculatorFind the area of a triangle with base 10 cm and height 6 cm.
- (1 mark)Grade 2G3Non-calculatorTwo angles of a triangle are 50° and 70°. Find the third.
- (1 mark)Grade 3G16Non-calculatorFind the volume of a cuboid 5 cm by 4 cm by 3 cm.
- (2 marks)Grade 3G3Non-calculatorFind the size of each interior angle of a regular hexagon.
Grades 4–5 · Foundation and Higher
- (2 marks)Grade 4G17Find the area of a circle of radius 6 cm, to 1 decimal place.
- (2 marks)Grade 4G20A right-angled triangle has shorter sides 5 cm and 12 cm. Find the hypotenuse.
- (2 marks)Grade 5G20A right-angled triangle has hypotenuse 10 cm and an angle of 35°. Find the side opposite the 35° angle, to 3 s.f.
- (2 marks)Grade 5G18Find the arc length of a sector with radius 8 cm and angle 45°. Give your answer to 3 s.f.
Grades 6–7 · Higher
- (2 marks)Grade 6G17A cone has radius 3 cm and height 7 cm. Find its volume, to 3 s.f. (V = ⅓πr²h)
- (2 marks)Grade 6G19Non-calculatorTwo similar shapes have lengths in the ratio 1 : 3. The smaller has area 8 cm². Find the area of the larger.
- (3 marks)Grade 7G22In triangle ABC, a = 8 cm, angle A = 40° and angle B = 65°. Find b, to 3 s.f.
- (2 marks)Grade 7G21Non-calculatorWithout a calculator, find the exact value of sin 30° + cos 60°.
Grades 8–9 · Higher
- (3 marks)Grade 8G22Two sides of a triangle are 7 cm and 9 cm, with an angle of 50° between them. Find the third side, to 3 s.f.
- (2 marks)Grade 8G23Non-calculatorFind the area of a triangle with sides 6 cm and 9 cm and an included angle of 30°.
- (3 marks)Grade 9G25Non-calculatorOA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.
- (4 marks)Grade 9G19Two similar solids have surface areas 50 cm² and 72 cm². The smaller has volume 250 cm³. Find the volume of the larger.
Worked solutions
Show worked solutions
Grade Ladder: Geometry and Measures — Worked Solutions
Full worked solutions to all 18 questions (36 marks).
Grades 1–3 · Foundation
- (1 mark)Grade 1G3Non-calculatorTwo angles lie on a straight line. One is 65°. Find the other.Solution
- Angles on a straight line add to 180°.
115° - (1 mark)Grade 1G17Non-calculatorFind the perimeter of a rectangle 7 cm by 4 cm.Solution
- 7 + 4 + 7 + 4.
22 cm - (1 mark)Grade 2G16Non-calculatorFind the area of a triangle with base 10 cm and height 6 cm.Solution
- ½ × 10 × 6.
30 cm² - (1 mark)Grade 2G3Non-calculatorTwo angles of a triangle are 50° and 70°. Find the third.Solution
- 180 − 50 − 70.
60° - (1 mark)Grade 3G16Non-calculatorFind the volume of a cuboid 5 cm by 4 cm by 3 cm.Solution
- 5 × 4 × 3.
60 cm³ - (2 marks)Grade 3G3Non-calculatorFind the size of each interior angle of a regular hexagon.Solution
- Exterior angle = 360 ÷ 6 = 60°.
- Interior = 180 − 60.
120°
Grades 4–5 · Foundation and Higher
- (2 marks)Grade 4G17Find the area of a circle of radius 6 cm, to 1 decimal place.Solution
- π × 6² = 36π.
113.1 cm² - (2 marks)Grade 4G20A right-angled triangle has shorter sides 5 cm and 12 cm. Find the hypotenuse.Solution
- √(5² + 12²) = √169.
13 cm - (2 marks)Grade 5G20A right-angled triangle has hypotenuse 10 cm and an angle of 35°. Find the side opposite the 35° angle, to 3 s.f.Solution
- opp = hyp × sin 35°.
5.74 cm - (2 marks)Grade 5G18Find the arc length of a sector with radius 8 cm and angle 45°. Give your answer to 3 s.f.Solution
- 45360 × 2π × 8 = 2π.
6.28 cm
Grades 6–7 · Higher
- (2 marks)Grade 6G17A cone has radius 3 cm and height 7 cm. Find its volume, to 3 s.f. (V = ⅓πr²h)Solution
- ⅓ × π × 9 × 7 = 21π.
66.0 cm³ - (2 marks)Grade 6G19Non-calculatorTwo similar shapes have lengths in the ratio 1 : 3. The smaller has area 8 cm². Find the area of the larger.Solution
- Area scale factor = 3² = 9.
- 8 × 9.
72 cm² - (3 marks)Grade 7G22In triangle ABC, a = 8 cm, angle A = 40° and angle B = 65°. Find b, to 3 s.f.Solution
- Sine rule: b/sin 65° = 8/sin 40°.
- b = 8 sin 65° ÷ sin 40° = 11.2797…
11.3 cm - (2 marks)Grade 7G21Non-calculatorWithout a calculator, find the exact value of sin 30° + cos 60°.Solution
- sin 30° = ½ and cos 60° = ½.
1
Grades 8–9 · Higher
- (3 marks)Grade 8G22Two sides of a triangle are 7 cm and 9 cm, with an angle of 50° between them. Find the third side, to 3 s.f.Solution
- Cosine rule: c² = 7² + 9² − 2 × 7 × 9 × cos 50°.
- c² = 49.0088…
7 cm - (2 marks)Grade 8G23Non-calculatorFind the area of a triangle with sides 6 cm and 9 cm and an included angle of 30°.Solution
- ½ × 6 × 9 × sin 30° = 27 × ½.
13.5 cm² - (3 marks)Grade 9G25Non-calculatorOA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.Solution
- AB = b − a.
- OM = OA + ½AB = a + ½(b − a).
½(a + b) - (4 marks)Grade 9G19Two similar solids have surface areas 50 cm² and 72 cm². The smaller has volume 250 cm³. Find the volume of the larger.Solution
- Area ratio 50 : 72 = 25 : 36, so length ratio 5 : 6.
- Volume ratio 125 : 216.
- 250 × 216125.
432 cm³
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