Mensuration: IGCSE Mathematics questions and answers
10 free IGCSE Mathematics Mensuration questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers Units of measure, Area & perimeter, Circles, arcs & sectors, Surface area & volume. Work through the questions below, then generate a full Mensuration paper with a fresh set every time.
A square ABCD has its side AB on the diameter of a semicircle of radius r, and its vertices C and D on the arc.
(a)
Find the area of the square in terms of r.
[4]
(b)
A second square is drawn inside a full circle of radius r with all four vertices on the circle. Find the ratio of the area of the square in the semicircle to the area of the square in the full circle, and determine which square fills the greater fraction of its own shape.
[6]
(c)
Find the largest possible area of a rectangle drawn in the semicircle with one side on the diameter. Justify that it is a maximum.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(s/2)² + s² = r² → s² = 4r²/5
O is the midpoint of AB, so OB = s/2 B1 OC = r and Pythagoras in triangle OBC M1 s24 + s2 = r2A1 s2 = 45r2A1
4
(b)
2 : 5; semicircle square fills 8/(5π) ≈ 0.509, circle square fills 2/π ≈ 0.637 — the circle square
diagonal of second square = 2r M1 area 2r2A1 ratio 45r2 : 2r2 = 2 : 5 A1 fraction of semicircle: 85πB1 fraction of circle: 2πB1 2π > 85π, so the square in the circle A1
6
(c)
r² (half-width r/√2)
area = 2x√(r2 − x2) with half-width x M1 maximise A2 = 4x2(r2 − x2), a quadratic in x2M1 x2 = r22 (completing the square / symmetry) A1 maximum area r2A1
4
Question 2Structured · 8 marksA* challenge
OAB is a sector of a circle with centre O, radius 6 cm and angle AOB = 120°.
(a)
Find the exact area of the segment between the chord AB and the arc AB. Give your answer in the form aπ − b√3.
[4]
(b)
Find the exact perimeter of the segment.
[3]
(c)
The segment is the cross-section of a prism of length 10 cm. Find the exact volume of the prism.
A solid metal cylinder has radius 6 cm and height 10 cm. It is melted down and made into solid spheres, each of radius 1.5 cm. Find the greatest number of complete spheres that can be made. [The volume, V, of a sphere with radius r is V = 43πr3.]
[4]
(b)
Calculate the volume of metal left over, as a percentage of the volume of the cylinder.
An equilateral triangle has sides of length 2 units. A circle is drawn inside it, touching all three sides.
(a)
Find the exact radius of the circle and the exact area of the region inside the triangle but outside the circle.
[5]
(b)
A smaller circle is drawn in one corner, touching the two sides at that corner and touching the first circle. Find its exact radius.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
r = 1/√3; area √3 − π/3
height of triangle √3 B1 centre is 13 of the height from the base (or 30° triangle) M1 r = √33 = 1√3A1 triangle area √3 B1 √3 − π3A1
5
(b)
√39 = 1/(3√3)
centres lie on the line from the vertex to the centre (angle bisector) M1 distance from vertex to a centre = 2 × radius (30° triangle) M1 with large radius R and small radius r: 2R − 2r = R + r A1 r = R3 = √39A1
4
Question 5Structured · 4 marks
(a)
Change 3.6 m² into cm².
[1]
(b)
Change 2750 cm³ into litres.
[1]
(c)
Change 54 km/h into m/s.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
36000 cm²
36000 B1
1
(b)
2.75 litres
2.75 B1
1
(c)
15 m/s
× 1000 ÷ 3600 M1 15 A1
2
Question 6Structured · 7 marks
The diagram shows a solid cylinder.
Fig. 6.1
(a)
Calculate the volume of the cylinder.
[2]
(b)
Calculate the total surface area of the cylinder.
[3]
(c)
The cylinder is made from metal of density 8.9 g/cm³. Calculate the mass of the cylinder in kilograms.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
2830 cm³
π × 62 × 25 M1 2830 A1
2
(b)
1170 cm²
2 × π × 62M1 2 × π × 6 × 25 M1 1170 A1
3
(c)
see working
their volume × density M1 ÷ 1000, correct answer A1
2
Question 7Structured · 5 marks
The diagram shows a sector of a circle.
Fig. 7.1
(a)
Calculate the area of the sector.
[2]
(b)
Calculate the perimeter of the sector.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
9.82 cm²
45360 × π × 52M1 9.82 A1
2
(b)
13.9 cm
45360 × 2 × π × 5 M1 + 2 × 5 M1 13.9 A1
3
Question 8Structured · 2 marks
A trapezium has parallel sides of 7 cm and 15 cm. The perpendicular distance between them is 5 cm. Calculate the area.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
55 cm²
12 × (7 + 15) × 5 M1 55 A1
2
Question 9Structured · 3 marks
A shape is made from a rectangle 10 cm by 11 cm with a right-angled triangle on top. The triangle has base 11 cm and height 6 cm. Find the total area of the shape and the length of the sloping side of the triangle.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
143 cm²; 12.5 cm
110 + 33 M1 143 cm² A1 √(112 + 62) = 12.5 B1
3
Question 10Structured · 2 marks
A prism has a cross-section that is a right-angled triangle with base 6 cm and height 6 cm. The prism is 14 cm long. Find its volume.
In the 0580 syllabus, Mensuration covers Units of measure, Area & perimeter, Circles, arcs & sectors, Surface area & volume.
Where can I get Mensuration questions with mark schemes?
This page has 10 Mensuration questions with answers. For more, the free generator on mentorriyaz.in builds a new Mensuration paper with a full mark scheme each time you click.
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Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in