Edexcel IGCSE Math A Higher 4.10 3d Shapes and Volume Questions

Use this 3D mensuration page to identify solids, count edges, calculate surface area or volume and handle compound solids with consistent units.

Exam points

Name common solids and count faces, edges or vertices from diagrams accurately.
Calculate surface area or volume of prisms, cylinders, cones, spheres and hemispheres.
Convert volume units or combine solids before using density, area or volume formulae.

Covered syllabus points

4.10.A Names of solids 4.10.B Faces, edges and vertices 4.10.C The surface area of simple shapes using the area 4.10.D The surface area of a cylinder 4.10.E The volume of prisms 4.10.F Between units of volume within the metric system 4.10.HA The surface area and volume of a sphere and a right

[Maximum number: 3]

The diagram shows a solid triangular prism.

Figure for Question 1 — Edexcel IGCSE Math A Higher

Work out the total surface area of the triangular prism.

Diagram NOT accurately drawn

\begin{aligned} 8\times6&=48\\ \frac12\times8\times6&=24\\ 15\times8&=120\\ 15\times6&=90\\ 15\times10&=150 \end{aligned}

\text{Surface area}=120+90+150+24+24=408

Final answer:

408

M1 for a correct method to find the areas of 2 different faces, not two triangles;
allow $8\times6$ as one area.
M1 for adding 4 or 5 values for area, at least 3 from a correct method.
A1 cao. SC B2 for an answer of 456 if no other marks awarded.

[Maximum number: 5]

Here are a solid sphere and a solid cylinder.

Figure for Question 2 — Edexcel IGCSE Math A Higher

The radius of the sphere is $r \mathrm{~cm}$ .
The radius of the cylinder is $r \mathrm{~cm}$ .
The height of the cylinder is $2 r \mathrm{~cm}$ .
The total surface area of the cylinder is $k \pi \mathrm{~cm}^{2}$

(a)

Find an expression for k in terms of r.

[ 2 ]

$2\pi r^2+2\pi r(2r)=6\pi r^2$

Final answer:
$6\pi r^2$

(b)

Show that the ratio
total surface area of the cylinder: total surface area of the sphere is the same as the ratio
volume of the cylinder: volume of the sphere

[ 3 ]

$\text{Surface area ratio}=6r^2:4r^2=3:2$ $\text{Volume ratio}=2r^3:\frac{4}{3}r^3$ =6:4=3:2

Shown.

M1 for a correct surface-area ratio.
M1 for a correct volume ratio.
A1 oe, for example ratios could be $\frac{3}{2}:1$ .

[Maximum number: 4]

The diagram shows a cuboid and a cylinder.

Figure for Question 3 — Edexcel IGCSE Math A Higher

Diagram NOT accurately drawn

The dimensions of the cuboid are $x \mathrm{~cm}$ by 12 cm by 5 cm .
The volume of the cuboid is $270 \mathrm{~cm}^{3}$
The radius of the cylinder is $x \mathrm{~cm}$ .
The height of the cylinder is $2 x \mathrm{~cm}$ .

(a)

Work out the volume of the cylinder.

Give your answer correct to the nearest whole number.

[ 3 ]

Radius:
$r=\frac{270}{12\times5}=4.5$

Volume:
$\pi r^{2}h=\pi\times4.5^{2}\times9 = 572.5\ldots$

Final answer:
573

A1 accepts 572 to 573.

(b)

Change $1 \mathrm{~m}^{3}$ to $\mathrm{cm}^{3}$

\mathrm{cm}^{3}

[ 1 ]

$1\text{ m}^{3}=(100\text{ cm})^{3}=1000000\text{ cm}^{3}$

Final answer:
1000000

Accept:
$10^{6}$
or one million.

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)