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GCSE Maths Higher: Geometry practice questions
Ten free GCSE Maths Higher tier questions on trigonometry, exact values, the cosine rule, circle theorems, spheres, arcs, vectors and the equation of a circle, each with a worked solution. Suitable for AQA, Edexcel and OCR.
Questions originales rédigées par les professeurs Studoli dans l’esprit de l’examen GCSE. Ce ne sont pas des sujets officiels.
Mode chronométré
10 questions en 15 minutes, une question à la fois. Vos réponses sont corrigées à la fin ou quand le temps est écoulé.
Notion: Trigonometry
1. A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the side opposite the 30° angle.
Correction
C. 5 cm
- sin 30° = opposite ÷ hypotenuse.
- Opposite = 10 × sin 30° = 10 × 0.5 = 5 cm.
Notion: Exact trig values
2. What is the exact value of cos 60°?
Correction
B. 1/2
- Learn the exact values: cos 60° = 1/2 and sin 60° = √3/2.
Notion: Area of a triangle
3. Two sides of a triangle are 8 cm and 10 cm, and the angle between them is 30°. Find its area.
Correction
A. 20 cm²
- Area = ½ab sin C.
- ½ × 8 × 10 × sin 30° = 40 × 0.5 = 20 cm².
Notion: Cosine rule
4. Two sides of a triangle are 5 cm and 7 cm, with 60° between them. Find the third side.
Correction
D. √39 cm (about 6.24 cm)
- Cosine rule: c² = a² + b² − 2ab cos C.
- c² = 25 + 49 − 2 × 5 × 7 × 0.5 = 74 − 35 = 39.
- c = √39 ≈ 6.24 cm.
Notion: Circle theorems
5. The angle at the centre of a circle is ___ the angle at the circumference standing on the same arc.
Correction
C. twice
- This is one of the circle theorems: the angle at the centre is twice the angle at the circumference.
Notion: Circle theorems
6. What is the angle in a semicircle?
Correction
B. 90°
- An angle drawn from the ends of a diameter to the circumference is always a right angle.
Notion: Volume
7. Find the volume of a sphere of radius 3 cm. Give your answer in terms of π.
Correction
A. 36π cm³
- Volume of a sphere = (4/3)πr³.
- (4/3) × π × 27 = 36π cm³.
Notion: Arc length
8. Find the arc length of a sector with radius 6 cm and angle 60°, in terms of π.
Correction
D. 2π cm
- Arc length = (angle/360) × 2πr.
- (60/360) × 2π × 6 = (1/6) × 12π = 2π cm.
Notion: Vectors
9. a = (3, −2) and b = (1, 4) as column vectors. Find 2a − b.
Correction
C. (5, −8)
- 2a = (6, −4).
- 2a − b = (6 − 1, −4 − 4) = (5, −8).
Notion: Equation of a circle
10. What is the equation of a circle with centre (0, 0) and radius 5?
Correction
B. x² + y² = 25
- A circle with centre the origin and radius r is x² + y² = r².
- r = 5, so x² + y² = 25.