What is covered in Grade 10 Mathematics Measurement?

Surface area and volume of right prisms, cylinders, pyramids, cones and spheres. Effect of scale factor k on area (k²) and volume (k³). Composite solids.

Is Grade 10 Measurement tested in Paper 1 or Paper 2?

Grade 10 Mathematics Measurement is assessed in Paper 2 of the end-of-year examinations, as per the CAPS Mathematics curriculum guidelines.

When is Measurement taught in Grade 10 Mathematics?

Measurement is taught in Term 4 of Grade 10 according to the CAPS Annual Teaching Plan (ATP) for Mathematics in South Africa.

Is Grade 10 Mathematics Measurement on the CAPS curriculum?

Yes. Measurement is part of the official CAPS Mathematics curriculum for Grade 10, Term 4 in South Africa. The content on MathSciBuddy follows the Annual Teaching Plan (ATP) set by the Department of Basic Education.

Is this Grade 10 Mathematics study material free?

Yes. All chapter notes, definitions, formulas, and worked examples on MathSciBuddy are completely free to read. No account or subscription is needed to access CAPS-aligned study notes.

Grade 10 Mathematics

Grade 10 · Term 4Mathematics

Measurement

We revise surface area and volume of right prisms and cylinders, then extend to spheres, right pyramids and right cones. We also explore what happens to surface area and volume when any dimension is multiplied by a constant factor k.

Week 1

11.1 Surface Area & Volume of 3D Solids

Revise surface area and volume of right prisms and cylinders
Calculate surface area and volume of spheres, right pyramids and right cones
Study the effect on volume and surface area when any dimension is multiplied by factor k
Solve problems involving composite solids

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Real-World Connection

A manufacturer designing a tin can needs surface area to know how much metal to buy, and volume to know how much product it holds. Getting these formulas right is worth millions in material savings across a production run of millions of units.

Volume of a right prism

V = A_{\text{base}} \times h

$A_{\text{base}}$ = area of the base (any shape); $h$ = perpendicular height

Surface area of a right prism

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

$P_{\text{base}}$ = perimeter of base; $h$ = height; 2 bases + lateral faces

Surface area of cylinder

SA = 2\pi r^2 + 2\pi r h

$r$ = radius, $h$ = height; two circular ends + curved surface

Volume of cylinder

V = \pi r^2 h

Special right prism with circular base: $A_{\text{base}} = \pi r^2$

ℹ️ Note

A cylinder is a right prism with a circular base. Substituting $A_{\text{base}}=\pi r^2$ and $P_{\text{base}}=2\pi r$ into the general right-prism formulas gives the cylinder formulas directly: $V=\pi r^2 h$ and $SA=2\pi r^2+2\pi r h$ . Similarly, a cone is a right pyramid with a circular base.

Surface area of sphere

SA = 4\pi r^2

$r$ = radius

Volume of sphere

V = \frac{4}{3}\pi r^3

$r$ = radius

Volume of a right pyramid

V = \frac{1}{3} A_{\text{base}} \times h

$A_{\text{base}}$ = area of base; $h$ = perpendicular height from base to apex

Surface area of a square pyramid

SA = b^2 + 2bl

$b$ = base edge length; $l$ = slant height of each triangular face; 1 square base + 4 triangular faces

Volume of cone

V = \frac{1}{3}\pi r^2 h

Right pyramid with circular base: $A_{\text{base}} = \pi r^2$

Slant height (cone or pyramid)

l = \sqrt{r^2 + h^2}

$l$ = slant height from Pythagoras; $r$ = distance from centre of base to midpoint of base edge (or radius for cone)

Surface area of cone

SA = \pi r^2 + \pi r l

Base circle + lateral surface; $l$ = slant height

Property / Rule

Effect of scale factor k

If every dimension of a solid is multiplied by k: all lengths scale by k, all areas (surface area) scale by k², all volumes scale by k³.

SA_{\text{new}} = k^2 \cdot SA_{\text{old}}\qquad V_{\text{new}} = k^3 \cdot V_{\text{old}}

Worked Examples

Worked Example

Triangular prism — surface area and volume

Problem

A right triangular prism has an equilateral triangle base with side 6 cm and prism height 10 cm. Find the total surface area and volume.

Worked Example

Cylinder surface area and volume

Problem

\pi

Worked Example

Composite solid — cone on cylinder

Problem

r=4

Worked Example

Square-based right pyramid

Problem

b=8

Worked Example

Effect of scale factor

Problem

k=2

Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q

L2 · Routine Procedures2 Q

L3 · Complex Procedures2 Q

L4 · Problem Solving2 Q

L1 · Knowledge1 mark

State the formula for the volume of a sphere.

L1 · Knowledge2 marks

Find the volume of a cube with side length 5 cm.

L2 · Routine Procedures3 marks

A square pyramid has base edge

b=6

cm and slant height

l=5

cm. Find the total surface area.

L2 · Routine Procedures3 marks

A cone has

r=3

cm and slant height

l=5

cm. Find total surface area.

L3 · Complex Procedures3 marks

A cylinder has volume

200\pi

cm³ and height 8 cm. Find the radius.

L3 · Complex Procedures3 marks

All dimensions of a rectangular box are tripled (k = 3). By what factor does the volume increase? By what factor does the surface area increase?

L4 · Problem Solving5 marks

A hemisphere (flat face down) is placed on top of a cylinder of the same radius

r=6

cm and cylinder height

h=10

cm. Find total volume.

L4 · Problem Solving5 marks

A sphere just fits inside a cube of side 8 cm. What percentage of the cube's volume does the sphere occupy? (Give answer to 1 decimal place.)

Number Patterns