AQA GCSE Maths · 3.4.2

Mensuration and calculation

50 maths questions with worked answers with answers for 3.4.2 Mensuration and calculation in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Perimeter · G17
A rectangle is 8 cm by 3 cm. Find its perimeter.
Answer:
Perimeter = 2 × (length + width).
2 × (8 + 3) = 22 cm.
Question 2 · Perimeter · G17
A square has side length 7 m. Find its perimeter.
Answer:
A square has four equal sides.
Perimeter = 4 × 7 = 28 m.
Question 3 · Perimeter · G17
A rectangle has perimeter 30 cm and width 5 cm. Find its length.
Answer:
Half the perimeter is length + width: 30 ÷ 2 = 15 cm.
Length = 15 − 5 = 10 cm.
Question 4 · Area · G17
Find the area of a 9 cm by 4 cm rectangle.
Answer:
Area = length × width.
9 × 4 = 36 cm².
Question 5 · Area · G16
Find the area of a triangle with base 10 cm and perpendicular height 7 cm.
Answer:
Triangle area = 12 × base × perpendicular height.
12 × 10 × 7 = 35 cm².
Question 6 · Area · G16
A parallelogram has base 8 m and perpendicular height 5 m. Find its area.
Answer:
Parallelogram area = base × perpendicular height.
8 × 5 = 40 m².
Question 7 · Area · G16
A trapezium has parallel sides 6 cm and 10 cm and perpendicular height 4 cm. Find its area.
Answer:
Trapezium area = 12 × (sum of parallel sides) × height.
12 × (6 + 10) × 4 = 32 cm².
Question 8 · Area · G17
A square has area 81 cm². Find its side length.
Answer:
Side length squared is 81.
The positive side length is √81 = 9 cm.
Question 9 · Area · G17
A rectangle is 12 m long with area 96 m². Find its width.
Answer:
Width = area ÷ length.
96 ÷ 12 = 8 m.
Question 10 · Circles · G17
A circle has radius 5 cm. Find its circumference in terms of π.
Answer:
Circumference = 2πr.
2π × 5 = 10π cm.
Question 11 · Circles · G17
A circle has diameter 14 cm. Find its circumference in terms of π.
Answer:
Circumference = π × diameter.
π × 14 = 14π cm.
Question 12 · Circles · G17
A circle has radius 4 cm. Find its area in terms of π.
Answer:
Area = πr².
π × 4² = 16π cm².
Question 13 · Circles · G17
A circle has area 25π cm². Find its radius.
Answer:
Area = πr², so πr² = 25π.
r² = 25; the positive radius is 5 cm.
Question 14 · Sectors · G18
Find the area of a 90° sector of a circle with radius 4 cm, in terms of π.
Answer:
Sector area = 90360 of the circle's area.
90360 × π × 4² = 4π cm².
Question 15 · Sectors · G18
Find the arc length of a 60° sector of radius 6 cm, in terms of π.
Answer:
Arc length = 60360 of the full circumference.
60360 × 2π × 6 = 2π cm.
Question 16 · Sectors · G18
Find the perimeter of a semicircle with radius 3 cm, in terms of π.
Answer:
The semicircular arc is half of 2πr: 3π cm.
Add the diameter 6 cm: perimeter = (3π + 6) cm.
Question 17 · Volume · G16
Find the volume of a cuboid 5 cm by 4 cm by 3 cm.
Answer:
Cuboid volume = length × width × height.
5 × 4 × 3 = 60 cm³.
Question 18 · Volume · G16
A prism has cross-sectional area 12 cm² and length 9 cm. Find its volume.
Answer:
Prism volume = cross-sectional area × length.
12 × 9 = 108 cm³.
Question 19 · Volume · G16
A cube has volume 64 cm³. Find its edge length.
Answer:
The edge cubed equals 64.
4³ = 64, so the edge is 4 cm.
Question 20 · Volume · G16
A triangular prism has triangular cross-sectional area 15 cm² and length 8 cm. Find its volume.
Answer:
Prism volume = cross-sectional area × length.
15 × 8 = 120 cm³.
Question 21 · Volume · G16
A cylinder has radius 3 cm and height 5 cm. Give its volume in terms of π.
Answer:
Cylinder volume = πr²h.
π × 3² × 5 = 45π cm³.
Question 22 · Volume · G17
A cone has radius 3 cm and perpendicular height 4 cm. Give its volume in terms of π.
Answer:
Cone volume = 13 × πr²h.
13 × π × 3² × 4 = 12π cm³.
Question 23 · Volume · G17
A sphere has radius 3 cm. Give its volume in terms of π.
Answer:
Sphere volume = 43 × πr³.
43 × π × 3³ = 36π cm³.
Question 24 · Volume · G17
A pyramid has base area 30 cm² and height 6 cm. Find its volume.
Answer:
Pyramid volume = 13 × base area × height.
13 × 30 × 6 = 60 cm³.
Question 25 · Surface area · G17
Find the total surface area of a cube with side 4 cm.
Answer:
There are 6 square faces, each with area 4² = 16 cm².
Total area = 6 × 16 = 96 cm².
Question 26 · Surface area · G17
Find the total surface area of a 2 cm by 3 cm by 4 cm cuboid.
Answer:
The three different face areas are 2 × 3 = 6, 2 × 4 = 8 and 3 × 4 = 12.
Each occurs twice: 2(6 + 8 + 12) = 52 cm².
Question 27 · Surface area · G17
A closed cylinder has radius 2 cm and height 5 cm. Give its total surface area in terms of π.
Answer:
Two circular ends have area 2πr² = 8π cm².
Curved area = 2πrh = 20π cm².
Total = 8π + 20π = 28π cm².
Question 28 · Pythagoras · G20
A right-angled triangle has legs 6 cm and 8 cm. Find its hypotenuse.
Answer:
Pythagoras: hypotenuse² = 6² + 8² = 36 + 64 = 100.
Hypotenuse = √100 = 10 cm.
Question 29 · Pythagoras · G20
A right-angled triangle has hypotenuse 13 cm and one leg 5 cm. Find the other leg.
Answer:
Pythagoras: other leg² = 13² − 5² = 169 − 25 = 144.
Other leg = √144 = 12 cm.
Question 30 · Pythagoras · G20
Find the diagonal of a 9 cm by 12 cm rectangle.
Answer:
Pythagoras: diagonal² = 9² + 12² = 81 + 144 = 225.
Diagonal = √225 = 15 cm.
Question 31 · Pythagoras · G20 · Higher only
A cuboid measures 2 cm by 3 cm by 6 cm. Find its space diagonal.
Answer:
Apply Pythagoras in three dimensions: diagonal² = 2² + 3² + 6² = 49.
Diagonal = √49 = 7 cm.
Question 32 · Trigonometry · G20
In a right-angled triangle, the side opposite angle θ is 3 and the hypotenuse is 6. Find sin θ.
Answer:
sin θ = opposite ÷ hypotenuse.
sin θ = 36 = 12.
Question 33 · Trigonometry · G20
In a right-angled triangle, the side adjacent to angle θ is 4 and the hypotenuse is 5. Find cos θ.
Answer:
cos θ = adjacent ÷ hypotenuse.
cos θ = 45.
Question 34 · Trigonometry · G20
In a right-angled triangle, the side opposite angle θ is 6 and the adjacent side is 8. Find tan θ.
Answer:
tan θ = opposite ÷ adjacent.
tan θ = 68 = 34.
Question 35 · Trigonometry · G21
State the exact value of sin 30°.
Answer:
In a 30°, 60°, 90° triangle, the side opposite 30° is half the hypotenuse.
sin 30° = 12.
Question 36 · Trigonometry · G21
State the exact value of cos 60°.
Answer:
In a 30°, 60°, 90° triangle, the side adjacent to 60° is half the hypotenuse.
cos 60° = 12.
Question 37 · Trigonometry · G21
State the exact value of tan 45°.
Answer:
In a 45°, 45°, 90° triangle, the two shorter sides are equal.
tan 45° = opposite ÷ adjacent = 1.
Question 38 · Trigonometry · G21
State the exact value of sin 45°.
Answer:
In a 45°, 45°, 90° triangle with shorter sides 1, the hypotenuse is √2.
sin 45° = 1√2 = √22.
Question 39 · Trigonometry · G21
State the exact value of cos 30°.
Answer:
A 30°, 60°, 90° triangle can have sides 1, √3 and 2.
cos 30° = adjacent ÷ hypotenuse = √32.
Question 40 · Trigonometry · G20
A right-angled triangle has hypotenuse 10 cm and an angle of 30°. Find the opposite side.
Answer:
sin 30° = opposite ÷ 10 = 12.
Opposite = 10 × 12 = 5 cm.
Question 41 · Trigonometry · G20
A right-angled triangle has adjacent side 4 cm and angle 45°. Find the opposite side.
Answer:
tan 45° = opposite ÷ 4 = 1.
Opposite = 4 × 1 = 4 cm.
Question 42 · Sine rule · G22 · Higher only
In triangle ABC, A = 30°, B = 90° and side a = 5 cm. Find side b.
Answer:
The sine rule gives bsin 90° = 5sin 30°.
sin 90° = 1 and sin 30° = 12.
Therefore b = 5 ÷ 12 = 10 cm.
Question 43 · Cosine rule · G22 · Higher only
A triangle has sides b = 3 cm and c = 4 cm with included angle A = 90°. Find side a.
Answer:
Cosine rule: a² = b² + c² − 2bc cos A.
a² = 3² + 4² − 2 × 3 × 4 × 0 = 25.
a = 5 cm.
Question 44 · Triangle area · G23 · Higher only
Two sides are 6 cm and 8 cm with included angle 30°. Find the area.
Answer:
Area = 12 × 6 × 8 × sin 30°.
sin 30° = 12, so the area is 12 cm².
Question 45 · Similarity · G19 · Higher only
Similar shapes have length scale factor 2. The smaller area is 7 cm². Find the larger area.
Answer:
Area scale factor = 2² = 4.
Larger area = 7 × 4 = 28 cm².
Question 46 · Similarity · G19 · Higher only
Similar solids have length scale factor 3. The smaller volume is 4 cm³. Find the larger volume.
Answer:
Volume scale factor = 3³ = 27.
Larger volume = 4 × 27 = 108 cm³.
Question 47 · Similarity · G19 · Higher only
The areas of two similar figures are 9 cm² and 36 cm². Find their length scale factor from small to large.
Answer:
Area scale factor = 36 ÷ 9 = 4.
Length scale factor = √4 = 2.
Question 48 · Units · G14
A rectangle is 2 m by 30 cm. Find its area in cm².
Answer:
Convert 2 m to 200 cm.
Area = 200 × 30 = 6,000 cm².
Question 49 · Bearings · G15
A bearing is 3°. Write it as a three-figure bearing.
Answer:
Bearings are written with three digits.
Add two leading zeros: 003°.
Question 50 · Bearings · G15
The bearing of B from A is 070°. Find the bearing of A from B.
Answer:
The reverse bearing differs by 180°.
070° + 180° = 250°.

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