Practice Maths

Volume of Cylinders and Composite Solids

Key Ideas

Key Terms

Volume
= base area × height (for any prism or cylinder).
Rectangular prism
V = lwh
Triangular prism
V = (½bhtriangle) × length
Cylinder
V = πr²h
Composite solid
Break the solid into simpler parts, calculate each volume, then add (or subtract for hollow sections).
Unit conversions
1 L = 1000 cm³  |  1 m³ = 1000 L  |  1 m³ = 1 000 000 cm³
Hot Tip When dealing with composite solids, always decide whether you are adding volumes (two solid pieces joined together) or subtracting volumes (a hole or hollow section cut out). Draw a diagram and label each part.

Worked Example

Question: Find the volume of a cylinder with radius r = 4 m and height h = 6 m. Give the exact answer and an approximation to 1 decimal place.

Step 1 — Identify formula:   V = πr²h

Step 2 — Substitute:   V = π × 4² × 6 = π × 16 × 6 = 96π m³

Step 3 — Approximate:   V = 96π ≈ 301.6 m³

Volume of a Cylinder

Volume measures the amount of space a solid occupies — or the amount of liquid it can hold. For a cylinder, you are essentially stacking circular "slices" of area πr2 through the full height h:

V = πr2h

where r is the radius of the circular base and h is the perpendicular height (not a slant height). Volume is measured in cubic units: cm3, m3, etc.

Real examples: a water tank, a drinking glass, a roll of cling wrap — all cylindrical volumes are found with this formula.

Composite Solids

A composite solid is a shape made by combining two or more simple solids. To find the total volume, break the solid into recognisable parts and handle each separately:

  • If the parts are joined together, add their volumes: Vtotal = V1 + V2.
  • If one part is removed from another (like a hole bored through a prism), subtract: Vtotal = Vouter − Vremoved.

Common composite combinations: a cylinder on top of a rectangular prism, a hemisphere on a cylinder, a cylinder with a cylindrical hole drilled through it.

Converting Volume Units

Knowing unit conversions is essential for real-world problems:

  • 1 cm3 = 1 mL (millilitre)
  • 1000 cm3 = 1 L (litre)
  • 1 m3 = 1 000 000 cm3 = 1000 L

For example, if a tank has a volume of 54 000 cm3, it holds 54 000 ÷ 1000 = 54 litres.

Setting Up Composite Volume Problems

For each composite solid problem: (1) sketch and label the solid, (2) draw dotted lines to show how you are splitting it, (3) identify the formula for each part, (4) calculate each volume, (5) add or subtract as required, (6) convert units if needed.

Key tip: Always check whether the question gives you the radius or the diameter. The formula V = πr2h requires the radius. If you are given the diameter, halve it before substituting. This is one of the most common errors in cylinder volume questions.

Mastery Practice

  1. Find the volume of each cylinder. Give exact answers in terms of π and decimal approximations to 1 decimal place. Fluency

    1. r = 5 cm, h = 8 cm
    2. r = 3 m, h = 10 m
    3. r = 6.5 cm, h = 4 cm
    4. diameter = 14 mm, h = 20 mm
    5. diameter = 1.2 m, h = 3.5 m

  2. Find the volume of each prism. Fluency

    1. Rectangular prism: l = 9 cm, w = 4 cm, h = 6 cm
    2. Rectangular prism: l = 2.5 m, w = 1.8 m, h = 1.2 m
    3. Triangular prism: triangle has base 8 cm and perpendicular height 5 cm; prism length = 12 cm
    4. Triangular prism: triangle has base 10 m and perpendicular height 6 m; prism length = 4 m

  3. Find the missing dimension. Understanding

    1. A cylinder has V = 200π cm³ and r = 5 cm. Find h.
    2. A cylinder has V = 1000 cm³ and h = 20 cm. Find r (to 2 d.p.).
    3. A rectangular prism has V = 360 cm³, l = 10 cm, w = 6 cm. Find h.
    4. A triangular prism has V = 180 m³. The triangular cross-section has base 9 m and perpendicular height 5 m. Find the prism length.

  4. Unit conversion and capacity. Understanding

    1. A rectangular tank has dimensions 80 cm × 60 cm × 50 cm. Find its volume in cm³, then convert to litres.
    2. A cylindrical water tank has r = 1.5 m and h = 2.8 m. Find the volume in m³, then convert to litres (1 m³ = 1000 L).
    3. A swimming pool holds 250 000 L of water. Express this volume in m³.
    4. A glass of water has a capacity of 250 mL. How many glasses can be filled from a 3-litre jug?

  5. A composite solid consists of a rectangular prism (10 cm × 6 cm × 4 cm) with a triangular prism on top. The triangular prism has the same 10 cm × 6 cm base, a triangular height of 3 cm, and runs the full 10 cm length. Understanding

    1. Find the volume of the rectangular prism.
    2. Find the volume of the triangular prism on top.
    3. Find the total volume of the composite solid.
    4. Sketch the solid and label all dimensions.

  6. A hollow cylindrical pipe has an outer radius of 5 cm, inner radius of 4 cm (wall thickness 1 cm), and length 2 m. Understanding

    1. Find the volume of the outer cylinder (using the outer radius). Give your answer in cm³.
    2. Find the volume of the hollow space inside the pipe.
    3. Hence find the volume of material in the pipe wall.
    4. If the pipe material has a density of 7.8 g/cm³, find the mass of the pipe in kg.

  7. A cylindrical fuel tank lies on its side. It has a diameter of 1.4 m and a length of 3 m. Problem Solving

    1. Find the volume of the tank in m³, to 3 d.p.
    2. Convert this to litres.
    3. Fuel costs $2.18 per litre. What is the value of a full tank, to the nearest dollar?
    4. If the tank is currently 40% full, how many litres of fuel are needed to fill it completely?

  8. A swimming pool has a rectangular section 15 m long and 8 m wide that is uniformly 1.2 m deep, plus a deeper rectangular section 5 m long and 8 m wide that is uniformly 2.4 m deep. Problem Solving

    1. Find the volume of the shallow section in m³.
    2. Find the volume of the deep section in m³.
    3. Find the total volume of the pool.
    4. How many litres of water does the pool hold?
    5. A pump can fill the pool at 800 litres per minute. How long (in hours and minutes) will it take to fill the pool from empty?

  9. A solid rectangular block of wood measures 20 cm × 12 cm × 8 cm. A cylindrical hole of radius 3 cm and depth 8 cm is drilled through the block from top to bottom. Problem Solving

    1. Find the volume of the original rectangular block.
    2. Find the volume of the cylindrical hole (to 2 d.p.).
    3. Find the volume of wood remaining after drilling.
    4. Wood has a density of 0.6 g/cm³. Find the mass of the drilled block in grams.

  10. A rainwater tank is in the shape of a cylinder with radius 0.9 m and height 1.8 m. Problem Solving

    1. Find the maximum capacity of the tank in m³ (to 3 d.p.).
    2. Convert this to litres.
    3. A household uses an average of 120 litres of tank water per day. How many full days will a full tank supply?
    4. After a rain event, 850 litres are collected. What fraction (as a percentage, to 1 d.p.) of the tank capacity is this?
    5. If the tank currently holds 1200 litres, how many more litres can it still accept?
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