Area & perimeter: IGCSE Mathematics questions with answers
Free Area & perimeter questions for IGCSE Mathematics (0580), part of Mensuration. Each question has the answer and the marking points an examiner would look for.
A badge is made from a rectangle 16 cm by 6 cm with a semicircle of diameter 6 cm joined to one of the 6 cm sides. A circular hole of radius 2 cm is cut out of the badge.
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 3Structured · 3 marks
A shape is made from a rectangle 16 cm by 12 cm with a right-angled triangle on top. The triangle has base 12 cm and height 3 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
210 cm²; 12.4 cm
192 + 18 M1 210 cm² A1 √(122 + 32) = 12.4 B1
3
Question 4Structured · 2 marks
A trapezium has parallel sides of length 5 cm and 13 cm. The perpendicular distance between them is 7 cm. Find its area.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
63 cm²
½(a + b)h M1 63 A1
2
Question 5Structured · 2 marks
A parallelogram has area 48 cm² and base 6 cm. Find its perpendicular height.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
8 cm
48 ÷ 6 M1 8 A1
2
Question 6Structured · 9 marksA* challenge
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
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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