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Similar shapes (area and vol) Quiz

Grade 6 • 6H • 10 Questions (33 Marks)

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Score: 0 / 10
Similar shapes (area and vol) — Notes & Worked Examples Show / Hide

Explanation:

If two similar shapes have length ratio a : b (scale factor k), then their area ratio is a2 : b2 (scale factor k2) and their volume ratio is a3 : b3 (scale factor k3).

Whenever you are given two areas or two volumes, always simplify the ratio first and square root (for area) or cube root (for volume) back to the length ratio a : b before finding any other measurement.

Example 1: Similar cylinders A and B have heights 4 cm and 10 cm. The surface area of A is 32 cm2. Find the surface area of B.

A 4 cm 32 cm² B 10 cm

Solution:

Length ratio = 4 : 10 = 2 : 5 → Area ratio = 22 : 52 = 4 : 25
Area of B = (32 ÷ 4) × 25 = 8 × 25 = 200 cm2
Answer: 200 cm2

Example 2: Similar solids P and Q have surface areas 27 cm2 and 75 cm2. The volume of Q is 250 cm3. Find the volume of P.

Solution:

Simplify area ratio P : Q = 27 : 75 = 9 : 25 (÷ 3)
Square root for length ratio = 3 : 5
Cube for volume ratio = 33 : 53 = 27 : 125
Volume of P = (250 ÷ 125) × 27 = 54 cm3
Answer: 54 cm3
Question 1
[1 mark]
Two mathematically similar shapes A and B have corresponding lengths in the ratio 3 : 5.
The area of shape A is 72 cm2. Work out the area of shape B.
cm²
Worked Solution & Answer
Area scale factor = (53)2 = 259, so AreaB = 72 × 259 = 200 cm2. B1
Answer: 200 cm2 cm²
Question 2
[1 mark]
Two mathematically similar shapes A and B have corresponding lengths in the ratio 2 : 3.
The area of shape A is 16 cm2. Work out the area of shape B.
cm²
Worked Solution & Answer
Area scale factor = (32)2 = 94, so AreaB = 16 × 94 = 36 cm2. B1
Answer: 36 cm2 cm²
Question 3
[3 marks]
Here are two similar solid shapes.
AB
surface area of shape **A** : surface area of shape **B** = 3 : 4
The volume of shape **B** is 10 cm3
Work out the volume of shape **A**.
Give your answer correct to 3 significant figures.
cm³
Worked Solution & Answer
Length ratio = √3 : √4 = √3 : 2. Volume ratio = (√3)³ : 2³ = 3√3 : 8. Volume of A = 10 × (√3 / 2)³ = 10 × 0.649519 = 6.50 cm³ (3 s.f.). M2 A1
Answer: 6.50 cm³
Question Paper (PDF) • Mark Scheme (PDF)
Question 4 IGCSE Jan 2021 1H Q20
[3 marks]
**A** and **B** are two similar solids.
ABDiagram NOTaccurately drawn
**A** has a volume of 1836 cm3
**B** has a volume of 4352 cm3
**B** has a total surface area of 1120 cm2
Work out the total surface area of **A**.
cm²
Worked Solution & Answer
Volume ratio A : B = 1836 : 4352 = 27 : 64 M1
Length ratio = ∛27 : ∛64 = 3 : 4. Area ratio = 3² : 4² = 9 : 16 M1
Surface area of A = 1120 × (9 / 16) = 630 cm². A1
Answer: 630 cm² cm²
Question Paper (PDF) • Mark Scheme (PDF)
Question 5
[4 marks]
The diagram shows two similar solid triangular prisms, **A** and **B**.
h cmPrism A43.74 cm28.1 cmPrism B
The volume of prism **A** is 58.806 cm3
The volume of prism **B** is 1587.762 cm3
The cross section of each prism is a right-angled triangle.
For prism **B**
[INDENT]the length of the base of the triangle is 8.1 cm
[INDENT]the area of the triangle is 43.74 cm2
The height of the triangle for prism **A** is h cm.
Work out the value of h.
h =
Worked Solution & Answer
(The volume of a prism is area of cross-section × length)
Volume scale factor = 1587.762 ÷ 58.806 = 27 M2
⇒ linear scale factor = ∛27 = 3. Height of B = (2 × 43.74) ÷ 8.1 = 10.8 cm M1
⇒ h = 10.8 ÷ 3 = 3.6 cm A1
Answer: h = 3.6
Question Paper (PDF) • Mark Scheme (PDF)
Question 6
[4 marks]
**A** and **B** are two similar cylindrical containers.
AB
the surface area of container **A** : the surface area of container **B** = 4 : 9
Tyler fills container **A** with water.
She then pours all the water into container **B**.
Tyler repeats this and stops when container **B** is full of water.
Work out the number of times that Tyler fills container **A** with water.
You must show all your working.
Worked Solution & Answer
Length ratio = √4 : √9 = 2 : 3 M2
Volume ratio = 2³ : 3³ = 8 : 27 M1
Number of fills = 27 ÷ 8 = 3.375, so Tyler fills container A 4 times. A1
Answer: 4
Question Paper (PDF) • Mark Scheme (PDF)
Question 7 IGCSE Jan 2023 2H Q19
[4 marks]
A and B are two similar vases.
ABDiagram NOTaccurately drawn
The vases are such that
surface area of vase B = 2564 × surface area of vase A
and that
volume of vase A - volume of vase B = 541.8 cm3
Calculate the volume of vase B
cm³
Worked Solution & Answer
Length scale factor B : A = √25 : √64 = 5 : 8. Volume ratio B : A = 5³ : 8³ = 125 : 512 M2
Difference in parts = 512 − 125 = 387 parts = 541.8 cm³ ⇒ 1 part = 541.8 ÷ 387 = 1.4 cm³ M1
Volume of vase B = 125 × 1.4 = 175 cm³. A1
Answer: 175 cm³
Question Paper (PDF) • Mark Scheme (PDF)
Question 8 IGCSE Jun 2024 1HR Q18
[4 marks]
The diagram shows two similar vases, A and B
A30 cmB12 cmDiagram NOTaccurately drawn
The height of vase A is 30 cm
The height of vase B is 12 cm
Given that
surface area of vase A - surface area of vase B = 178.5 cm2
find the surface area of vase A
cm²
Worked Solution & Answer
Length ratio A : B = 30 : 12 = 5 : 2 ⇒ Area ratio A : B = 25 : 4 M2
25 − 4 = 21 parts = 178.5 cm² ⇒ 1 part = 178.5 ÷ 21 = 8.5 cm² M1
Surface area of vase A = 25 × 8.5 = 212.5 cm² A1
Answer: 212.5 cm²
Question Paper (PDF) • Mark Scheme (PDF)
Question 9 AQA Nov 2019 2H Q23
[4 marks]
Here are three similar cuboids, A, B and C.
A has length 5 cm, width 2 cm and height 3 cm
B has length 10 cm
C has length x cm
A5 cm2 cm3 cmB10 cmCx cm
(a) The total surface area of A is 62 cm2
Tim wants to work out the total surface area of B.
Here is his working:
10 ÷ 5 = 2
62 × 2 = 124
Total surface area of B = 124 cm2
Make one criticism of Tim's method. [1 mark]
(b) Volume of A × 1258 = Volume of C
Work out the value of x. [3 marks]
Answer
Worked Solution & Answer
(a)
The linear scale factor is 10 ÷ 5 = 2
Surface area scales with the square of the linear scale factor, so the area scale factor is 22 = 4, not 2. C1
Answer (a): He should have multiplied by 2² (4), not 2
(b)
Volume scale factor from A to C = 1258 M1
Linear scale factor = 3√(<span class="opg-frac"><span class="opg-num">125</span><span class="opg-den">8</span></span>) = 52 = 2.5 M1
Length of A = 5 cm.
x = 5 × 2.5 = 12.5 cm. A1
Answer (b): Answer 12.5
Question Paper (PDF) • Mark Scheme (PDF)
Question 10
[5 marks]
Two jars are mathematically similar.
90 cm³h cm1350 cm³12 cm
(a) Work out the height, h cm, of the smaller jar.
Give your answer correct to 1 decimal place. [4 marks]
cm
(b) Find the exact value of p. [1 mark]
p =
Worked Solution & Answer
(a)
Volume scale factor = 1350 ÷ 90 = 15 M3
Linear scale factor = ∛15 ≈ 2.4662. h = 12 ÷ ∛15 = 4.865... ≈ 4.9 cm A1
Answer (a): 4.9 cm
(b)
Area scale factor p = (linear scale factor)² = (∛15)² = 15 {}2/3) = ∛225 B1
Answer (b): p = 152/3 (or 3√(225))
Question Paper (PDF) • Mark Scheme (PDF)
Score: 0 / 10 (0 / 33 marks)
Similar shapes (area and vol) Worksheets Next: Negative scale factors →

Grade 6 Quizzes

14 Topics
6A - Recurring decimals6B - Harder indices
6C - The product rule6D - Repeated change
6E - Triple brackets6F - Parallel and perpendicular
6G - Inequalities on graphs6H - Similar shapes (area and vol)
6J - Negative scale factors6K - Circle theorems
6L - Cumulative frequency6M - Box plots
6N - Combined transformations6P - Capture recapture