Surface Area & Volume of 3-D Objects — Grade 9 Maths (CAPS)

Volume and surface area of cubes, rectangular prisms, triangular prisms and cylinders, SI unit conversions, and what happens to volume when you double the dimensions — with drawn solids, worked examples and a quiz.

Also known as: surface area of a cylinder · volume of a prism

Grade 8 covered cubes, rectangular prisms and triangular prisms. Grade 9 adds the cylinder and takes the triangular prism further — the two shapes that appear most often in real life and in exams.

1The one idea behind every prism

Volume of any prism = area of the cross-section × length

A prism is a solid with the same shape all the way through. Work out the area of that flat cross-section, then multiply by how long the prism is. Every volume formula below is just this idea applied to a different cross-section.

2Rectangular prism (box) and cube

V = l × b × h · Surface area = 2(lb + lh + bh)

Worked ExampleA box measures 6 cm by 4 cm by 3 cm. Find its volume and surface area

1
V = 6 × 4 × 3 = 72 cm³.
Multiply the three dimensions.
2
SA = 2(6 × 4 + 6 × 3 + 4 × 3) = 2(24 + 18 + 12).
Three pairs of identical faces.
3
SA = 2(54) = 108 cm².
Area is in square units, volume in cubic units.

3Triangular prism

A triangular prism: find the area of the triangular cross-section, then multiply by the length.

Worked ExampleTriangular prism: base 8 cm, height 5 cm, length 10 cm. Find the volume

1
Cross-section area = ½ × 8 × 5 = 20 cm².
Area of the triangle.
2
V = 20 × 10.
Multiply by the length of the prism.
3
V = 200 cm³.

Triangular prism
V = area of the triangle × length
SA = 2 × area of the triangle + (perimeter of the triangle) × length

The surface is two identical triangular ends plus three rectangles. Each rectangle is as long as the prism and as wide as one side of the triangle — so adding the three sides FIRST (the perimeter) and multiplying once does all three at a stroke. That means you need all three sides of the triangle, and often one of them has to be found with Pythagoras.

Worked ExampleThe prism below has a right-angled cross-section with sides of 12 cm and 9 cm, and it is 10 cm long. Find its total surface area.

1
Third side: 12² + 9² = 144 + 81 = 225, so the slanted side is √225 = 15 cm.
Pythagoras — the perimeter needs all three sides.
2
The two triangular ends: 2 × ½ × 12 × 9 = 108 cm².
The ends are identical, so work out one and double it.
3
The three rectangles: (12 + 9 + 15) × 10 = 36 × 10 = 360 cm².
Perimeter of the triangle × length of the prism.
4
SA = 108 + 360 = 468 cm².
Ends plus sides.

⚠️

Using only the two sides you were given. A triangular prism has three rectangular faces, so the third side of the triangle must be found before you can work out the surface area — even though the volume never needs it.

4Cylinder

A cylinder: the cross-section is a circle of radius r, and the solid is h tall.

V = πr²h · Surface area = 2πr² + 2πrh

The volume is the circle's area (πr²) times the height. The surface area is the two circular ends (2πr²) plus the curved side, which unrolls into a rectangle of width 2πr and height h.

Worked ExampleA cylinder has radius 5 cm and height 10 cm. Find V and SA (π ≈ 3,14)

1
V = πr²h = 3,14 × 5² × 10.
Square the radius first.
2
V = 3,14 × 25 × 10 = 785 cm³.
3
SA = 2πr² + 2πrh = 2(3,14)(25) + 2(3,14)(5)(10).
Two ends plus the curved surface.
4
SA = 157 + 314 = 471 cm².

⚠️

Do not confuse πr² (area of the circular end) with 2πr (its circumference). Squaring the radius is the step learners forget most often.

5SI unit conversions

1 cm³ = 1 mℓ and 1 000 cm³ = 1 ℓ
1 m³ = 1 000 ℓ = 1 kℓ
Area: 1 cm² = 100 mm², 1 m² = 10 000 cm²

6What doubling does to the volume

This is a favourite exam question, and the answer surprises people.

Worked ExampleA cylinder's radius is doubled. What happens to the volume?

1
Original: V = πr²h.
2
New radius 2r: V = π(2r)²h = π(4r²)h.
Squaring 2r gives 4r² — the 2 gets squared too.
3
The volume becomes 4 times as big.
Doubling the radius multiplies volume by 2² = 4.

Double the height only → volume × 2
Double the radius only → volume × 4 (because r is squared)
Double every dimension → volume × 8

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
A cube has a volume of 512 cm³. Determine the length of one edge, and hence the total surface area of the cube.
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edge = ∛512 = 8 cm (2)
A cube has 6 square faces, so SA = 6 × 8² (1)
= 384 cm² (1)
2
Volume of a box 6 cm × 4 cm × 3 cm.
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72 cm³
3
Surface area of a box 6 cm × 4 cm × 3 cm.
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2(24 + 18 + 12) = 108 cm²
4
Volume of a cube with side 5 cm.
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5³ = 125 cm³
5
Triangular prism: base 6, height 4, length 12 cm. Find the volume.
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½(6)(4) = 12; 12 × 12 = 144 cm³
6
The diagram below shows a triangular prism whose cross-section is a right-angled triangle with sides of 15 cm and 8 cm. The prism is 5 cm long. Calculate the length of the third side of the triangle, and then the total surface area of the prism.
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Third side = √(15² + 8²) = √289 = 17 cm (2)
Two ends = 2 × ½ × 15 × 8 = 120 cm² (1)
Three rectangles = (15 + 8 + 17) × 5 = 200 cm² (1)
SA = 120 + 200 = 320 cm² (1)
7
Cylinder radius 3 cm, height 7 cm. Find the volume (π ≈ 3,14).
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3,14 × 9 × 7 = 197,82 cm³
8
Cylinder radius 3 cm, height 7 cm. Find the surface area (π ≈ 3,14).
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2(3,14)(9) + 2(3,14)(3)(7) = 56,52 + 131,88 = 188,4 cm²
9
Cylinder radius 2 cm, height 5 cm. Find the volume (π ≈ 3,14).
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3,14 × 4 × 5 = 62,8 cm³
10
How many mℓ does a container of 250 cm³ hold?
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250 mℓ (1 cm³ = 1 mℓ)
11
A cylinder's height is doubled. What happens to its volume?
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It doubles (height is not squared).
12
A cylinder's radius is doubled. What happens to its volume?
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It becomes 4 times bigger (r is squared).