Similarity: IGCSE Mathematics questions with answers
Free Similarity questions for IGCSE Mathematics (0580), part of Geometry. Each question has the answer and the marking points an examiner would look for.
Two bottles are mathematically similar. The surface areas of the bottles are 36 cm² and 225 cm². The smaller bottle has a volume of 64 cm³. Calculate the volume of the larger bottle.
[3]
(b)
A third bottle, also similar, has a volume of 512 cm³. Find the surface area of this bottle.
Triangle ABC is similar to triangle PQR. AB = 6 cm, BC = 4 cm and PQ = 15 cm.
(a)
Calculate QR.
[2]
(b)
The area of triangle ABC is 23 cm². Calculate the area of triangle PQR.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
10 cm
scale factor 2.5 M1 10 A1
2
(b)
143.75 cm²
× 2.52M1 143.75 A1
2
Question 3Structured · 7 marksA* challenge
In triangle ABC, D is the point on AB with AD : DB = 1 : 2 and E is the point on AC with AE : EC = 1 : 2. The lines BE and CD meet at X.
(a)
Explain why DE is parallel to BC and find the ratio DE : BC.
[2]
(b)
Find the fraction of the area of triangle ABC that lies inside triangle XBC.
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
triangle ADE is an enlargement of ABC, scale factor 1/3; DE : BC = 1 : 3
ADE similar to ABC (common angle, sides in ratio 1 : 3) M1 so DE ∥ BC and DE : BC = 1 : 3 A1
2
(b)
12
triangles XDE and XCB are similar, ratio 1 : 3 M1 so EX : XB = 1 : 3 A1 area EBC = 23 area ABC (same height, base EC = 23AC) M1 area XBC = 34 area EBC M1 12 (allow any equivalent route, e.g. DX : XC = 1 : 3) A1
5
Question 4Structured · 9 marksA* challenge
A solid cone has base radius 12 cm and height 36 cm. A smaller cone of height 12 cm is cut off the top by a cut parallel to the base, leaving a frustum. [Volume of a cone = ⅓πr²h]
(a)
Calculate the volume of the frustum.
[4]
(b)
Show that the volume of the frustum is 2627 of the volume of the original cone.
[2]
(c)
A mathematically similar frustum has a total surface area 4 times as large. It is made of a material with density 1.2 g/cm³. Calculate its mass in kilograms.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
5230 cm³
small cone is similar: radius 4 cm M1 large cone 5429 M1 small cone 201.1 M1 5230 cm³ A1
In triangle ABC, D is a point on AB and E is a point on AC. DE is parallel to BC. AD = 8 cm, DB = 5 cm, DE = 6 cm and AE = 4 cm.
(a)
Calculate BC.
[2]
(b)
Calculate EC.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
9.75 cm
scale factor 138M1 9.75 A1
2
(b)
2.5 cm
AC = 4 × 138 = 6.5 M1 2.5 A1
2
Question 6Structured · 5 marksA* challenge
(a)
Two bottles are mathematically similar. The surface areas of the bottles are 54 cm² and 150 cm². The smaller bottle has a volume of 108 cm³. Calculate the volume of the larger bottle.
[3]
(b)
A third bottle, also similar, has a volume of 864 cm³. Find the surface area of this bottle.
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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