Surface area and volume of composite objects

What you will learn

calculate the surface area of composite objects built from prisms and cylinders,
calculate the volume of composite objects by adding or subtracting component volumes,
identify shared (hidden) faces when computing exposed surface area,
solve practical problems involving rainwater tanks, packaging, and construction.

Worked example 0 Real-world example: rainwater tank capacity

A rainwater tank consists of a cylindrical body with radius $0.6$ m and height $1.5$ m, sitting on a rectangular concrete slab $1.4$ m by $1.4$ m by $0.1$ m thick. Find (a) the total volume of the assembly, and (b) the exposed surface area to be painted (the curved surface and top of the cylinder only — the slab is unpainted).

Cylinder volume: $V = \pi(0.6)^2(1.5) = 0.54\pi \approx 1.696$ m $^3$ (capacity $\approx 1696$ L).
Slab volume: $1.4 \times 1.4 \times 0.1 = 0.196$ m $^3$ .
Total volume $\approx 1.696 + 0.196 = 1.892$ m $^3$ .
Curved surface area of cylinder: $2\pi(0.6)(1.5) = 1.8\pi \approx 5.655$ m $^2$ .
Top circle: $\pi(0.6)^2 = 0.36\pi \approx 1.131$ m $^2$ .
Area to paint $\approx 5.655 + 1.131 = 6.786$ m $^2$ .

Key idea: always identify which faces are exposed (visible) and which are hidden where components join.

1. Strategy for composite objects

Every composite-object problem follows the same two-step strategy:

Volume: add (or subtract) the volumes of each component.

Surface area: add the surface areas of each component, then subtract the hidden (shared) faces — those internal areas where the shapes meet.

Composite volume

V_{\text{composite}} = V_1 + V_2 + \cdots \quad\text{(or subtract if a piece is removed)}

Composite surface area

\text{SA}_{\text{composite}} = \text{SA}_1 + \text{SA}_2 - 2 \times A_{\text{shared}}

The factor of $2$ appears because the shared face is counted once in each component’s full SA.

Worked example 1 Cylinder sitting on a prism

A cylinder of radius $3$ cm and height $5$ cm sits centred on top of a rectangular prism $10 \times 10 \times 4$ cm. Find the total exposed surface area.

Prism full SA: $2(100) + 4(40) = 200 + 160 = 360$ cm $^2$ .
Cylinder full SA: $2\pi(9) + 2\pi(3)(5) = 18\pi + 30\pi = 48\pi \approx 150.80$ cm $^2$ .
Shared face (circle where cylinder meets prism top): $A = \pi(3)^2 = 9\pi \approx 28.27$ cm $^2$ .
Subtract shared face twice (once from prism top, once from cylinder base): exposed SA $\approx 360 + 150.80 - 2(28.27) \approx 454.26$ cm $^2$ .

2. Volume of composite objects

Worked example 2 Prism with a cylindrical hole

A rectangular block of steel is $20 \times 12 \times 8$ cm. A cylindrical hole of radius $2$ cm is drilled all the way through the $20$ cm length. Find the remaining volume.

Block volume: $20 \times 12 \times 8 = 1920$ cm $^3$ .
Cylinder removed: $\pi(2)^2(20) = 80\pi \approx 251.33$ cm $^3$ .
Remaining volume: $1920 - 251.33 \approx 1668.67$ cm $^3$ .

Worked example 3 L-shaped prism

An L-shaped concrete pad can be split into two rectangular prisms: one is $4 \times 2 \times 0.3$ m and the other is $3 \times 2 \times 0.3$ m (with a $1 \times 2$ overlap removed). Total volume:

Think of the L as a full $4 \times 3 \times 0.3$ rectangle minus a $1 \times 1 \times 0.3$ corner block.
Full rectangle: $4 \times 3 \times 0.3 = 3.6$ m $^3$ .
Corner removed: $1 \times 1 \times 0.3 = 0.3$ m $^3$ .
Volume $= 3.6 - 0.3 = 3.3$ m $^3$ .

3. Surface area of composite objects

Worked example 4 Half-cylinder on a prism (shed roof)

A garden shed has a rectangular base $3$ m $\times$ $4$ m $\times$ $2$ m high, with a half-cylinder roof of radius $1.5$ m running along the $4$ m length. Find the total exposed surface area.

Prism without its top face: base $= 12$ , two long sides $= 2(4 \times 2) = 16$ , two short sides $= 2(3 \times 2) = 12$ . Total $= 12 + 16 + 12 = 40$ m $^2$ . (The top is covered by the roof.)
Half-cylinder curved surface: $\frac{1}{2}(2\pi)(1.5)(4) = 6\pi \approx 18.85$ m $^2$ .
Two semicircular ends: $2 \times \frac{1}{2}\pi(1.5)^2 = 2.25\pi \approx 7.07$ m $^2$ .
Total exposed SA $\approx 40 + 18.85 + 7.07 = 65.92$ m $^2$ .

Note: the top rectangle of the prism and the flat face of the half-cylinder cancel each other, so neither appears in the final count.

Worked example 5 Construction: concrete column and base

A square concrete base $0.8 \times 0.8 \times 0.2$ m supports a cylindrical column of radius $0.2$ m and height $3$ m. Find the total volume of concrete.

Base volume: $0.8 \times 0.8 \times 0.2 = 0.128$ m $^3$ .
Column volume: $\pi(0.2)^2(3) = 0.12\pi \approx 0.377$ m $^3$ .
Total $\approx 0.128 + 0.377 = 0.505$ m $^3$ .

Key formulas

Rectangular prism volume

V = l \times w \times h

Cylinder volume

V = \pi r^2 h

Cylinder surface area

\text{SA} = 2\pi r^2 + 2\pi r h

Prism surface area

\text{SA} = 2A_{\text{base}} + P_{\text{base}} \times L

Practice

Fluency

Tier 1: basic calculations

A cylinder ( $r = 4$ cm, $h = 10$ cm) sits on top of a cube of side $10$ cm. Find the total volume.
A rectangular prism $8 \times 6 \times 5$ cm has a cylindrical hole ( $r = 2$ cm) drilled through the $5$ cm height. Find the remaining volume.
Two rectangular prisms are joined end-to-end: one is $6 \times 4 \times 3$ cm, the other is $8 \times 4 \times 3$ cm. Find the total volume and the exposed surface area.
A cylinder ( $r = 3$ cm, $h = 7$ cm) has a hemisphere ( $r = 3$ cm) on top. Find the total volume.
Find the exposed surface area in question 4. (Hemisphere curved SA $= 2\pi r^2$ .)
An L-shaped block is formed from a $10 \times 6 \times 4$ cm prism with a $4 \times 3 \times 4$ cm block removed from one corner. Find the volume.
A half-cylinder ( $r = 5$ cm, $l = 12$ cm) sits on top of a rectangular prism $10 \times 12 \times 6$ cm. Find the total volume.
Find the total surface area for the solid in question 1, given that the cylinder sits centred on the top face of the cube.

Reasoning

Tier 2: mixed practice

A cylindrical water tank ( $r = 0.5$ m, $h = 1.8$ m) needs to be insulated on the curved surface and top only. Insulation costs $18 per m $^2$ . Find the total cost.
A swimming pool has a uniform rectangular cross-section $10 \times 5$ m. It is $1.2$ m deep at one end and $2.4$ m at the other (the floor slopes uniformly). Find the volume of water when the pool is full.
A factory chimney consists of a rectangular base $2 \times 2 \times 1$ m topped by a cylinder of radius $0.4$ m and height $8$ m. Find (a) the total volume, and (b) the total exposed surface area (the chimney is open at the top).
A packing box $30 \times 20 \times 15$ cm contains a cylindrical can ( $r = 5$ cm, $h = 15$ cm) standing upright. What percentage of the box volume is wasted space?
Two cylinders are joined: a large one ( $r = 6$ cm, $h = 10$ cm) with a smaller one ( $r = 3$ cm, $h = 8$ cm) centred on top. Find the total exposed surface area.
A solid is made by cutting a hemisphere ( $r = 4$ cm) from the top of a cylinder ( $r = 4$ cm, $h = 10$ cm). Find the remaining volume.

Reasoning

Tier 3: explain and apply

Explain why you must subtract the shared face area twice (not once) when finding the exposed surface area of two joined solids.
A composite tank is a cylinder ( $r = 1$ m, $h = 2$ m) with a cone on top (same radius, height $0.5$ m). The cone volume is $\frac{1}{3}\pi r^2 h$ . Find the total capacity in litres and explain why the cone adds relatively little capacity.
A manufacturer needs a container with volume $2000$ cm $^3$ . Design A is a single cylinder; Design B is a cube with a hemisphere on top. For each, find dimensions that achieve the target volume and compare total surface areas to determine which uses less material.
A rectangular prism $a \times a \times 2a$ has a cylinder of radius $\frac{a}{4}$ drilled through its longest dimension. Express the remaining volume as a function of $a$ .

Challenge

Reasoning

Harder reasoning

A silo consists of a cylinder of radius $3$ m and height $10$ m topped by a hemisphere. Find (a) the total volume, and (b) the total external surface area. If grain fills the silo to $\frac{3}{4}$ of its capacity, find the volume of grain.
A trophy is made from a rectangular prism base $8 \times 8 \times 2$ cm, with a cylinder ( $r = 2$ cm, $h = 12$ cm) rising from its centre, and a solid sphere ( $r = 3$ cm) on top of the cylinder. Find the total volume and the total exposed surface area. (Sphere SA $= 4\pi r^2$ ; sphere $V = \frac{4}{3}\pi r^3$ .)
An underground pipe is a hollow cylinder with outer radius $15$ cm and inner radius $12$ cm, running for $50$ m. Find the volume of material in the pipe wall.
A composite solid is formed by attaching a square-based pyramid (base $6 \times 6$ cm, slant height $5$ cm) to the top of a cube of side $6$ cm. Find the total exposed surface area. (Lateral area of a pyramid $= \frac{1}{2} \times \text{perimeter} \times \text{slant height}$ .)