8 min readConcept sheetA 3-mark question; often half of a 5-mark question

Volume of a combination of two solids

Volume does not hide anything. Add the volumes of the joined parts. If a part is cut out, subtract its volume from the bigger solid.

Joined solidsV = V₁ + V₂

Part removedV = V(whole) − V(removed part)

Cone of height H on a cylinder of height h (same r)V = πr²h + (1/3)πr²H = πr²(h + H/3)

Units1000 cm³ = 1 litre; 1 m³ = 1000 litres

Surface area or volume: what changes when you join?

Surface areaFaces that touch are hidden. So you do not add the full TSAs. Add only the exposed surfaces.

VolumeNo space is lost when solids are joined. So you simply add the volumes. When a part is cut out, subtract it.

Take a cone of height H on a cylinder of height h, both of radius r. Surface area = πrl + 2πrh + πr², but volume = πr²h + (1/3)πr²H.

Worked example 1: a cone on a cylinder

Problem

A solid is a cylinder of radius 3.5 cm and height 10 cm with a cone of the same radius and height 6 cm on top. Find its volume. (Use π = 22/7.)

Working

Given: r = 3.5 cm, cylinder height h = 10 cm, cone height H = 6 cm.
Formula: V = volume of cylinder + volume of cone = πr²h + (1/3)πr²H = πr²(h + H/3).
πr² = (22/7) × 3.5 × 3.5 = 38.5 cm².
h + H/3 = 10 + 6/3 = 10 + 2 = 12 cm.
V = 38.5 × 12 = 462 cm³.

Answer

Volume of the solid = 462 cm³.

Worked example 2: wood in a pen stand

Problem

A wooden pen stand is a cuboid 12 cm × 7 cm × 6 cm. Two cylindrical holes, each of radius 1.4 cm and depth 5 cm, are drilled in it for pens. Find the volume of wood left. (Use π = 22/7.)

Working

Formula: wood left = volume of cuboid − volume of the two holes.
Volume of cuboid = 12 × 7 × 6 = 504 cm³.
Volume of one hole = πr²h = (22/7) × 1.4 × 1.4 × 5.
(22/7) × 1.96 = 6.16, so one hole = 6.16 × 5 = 30.8 cm³.
Two holes = 2 × 30.8 = 61.6 cm³.
Wood left = 504 − 61.6 = 442.4 cm³.

Answer

Volume of wood left = 442.4 cm³.

Worked example 3: a medicine capsule

Problem

A capsule is a cylinder with a hemisphere at each end. Its total length is 14 mm and its diameter is 6 mm. Find its volume. (Use π = 3.14.)

Working

Given: r = 6 ÷ 2 = 3 mm. Each hemisphere is 3 mm long.
Length of the cylinder h = 14 − 3 − 3 = 8 mm.
Two hemispheres make one sphere. Formula: V = πr²h + (4/3)πr³.
V = π × 9 × 8 + (4/3) × π × 27 = 72π + 36π = 108π.
V = 108 × 3.14 = 339.12 mm³.

Answer

Volume of the capsule = 339.12 mm³ (about 0.34 cm³).

Steps for any volume of a combination

Name the parts, for example 'cone + hemisphere'.
Find every missing length: cone height = total height − hemisphere radius.
Write V = V₁ + V₂, or V = V(whole) − V(removed).
Take common factors like πr² outside a bracket to keep the arithmetic short.
Give the answer in cm³ or m³, and in litres if the question asks how much it holds.

In real life

A kulfi with a rounded top, an overhead water tank with a domed lid and a temple kalasam are combinations of solids. Their volume tells how much milk, water or metal they hold or use.

Exam angle

2 marks: “A solid is a hemisphere of radius 3 cm with a cone of the same radius and height 4 cm on it. Find its volume in terms of π.” V = (2/3)πr³ + (1/3)πr²h = (2/3)π × 27 + (1/3)π × 9 × 4 = 18π + 12π = 30π cm³.

3 marks: “A wooden cuboid 10 cm × 8 cm × 5 cm has a hemispherical hollow of radius 2.1 cm scooped out of its top. Find the volume of wood left. (Use π = 22/7.)” V = 10 × 8 × 5 − (2/3)πr³ = 400 − (2/3) × (22/7) × 9.261 = 400 − 19.404 = 380.596 cm³ (about 380.6 cm³).

5 marks: “A solid is a cylinder with a hemisphere at each end. Its total length is 17 cm and its diameter is 7 cm. Find its volume and its surface area. (Use π = 22/7.)” r = 3.5 cm, cylinder length h = 17 − 7 = 10 cm. V = πr²h + (4/3)πr³ = 385 + 179.67 = 564.67 cm³. Surface area = 2πrh + 4πr² = 2πr(h + 2r) = 22 × 17 = 374 cm².

Remember it

Area hides the joint; volume does not. Add the volumes, subtract the holes.