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Cambridge IGCSE Maths (0580) · Mensuration

Mini-Lesson

Mensuration

2 missions. Guess, learn, then check. Your place is saved.

Mission 1Why do runners in outer lanes start further ahead?
Mission 2Roll a sheet into a tall tube or a wide one. Which holds more?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

Why do runners in outer lanes start further ahead?

Make a guess: Why do runners in outer lanes start further ahead?

Measures · perimeter and area

Areas of triangles, parallelograms and trapezia

Perimeter is the total distance round the edge (cm). Area is the space inside (cm²).
Always use the perpendicular height, measured at 90° to the base, never a slanted side.
Split compound shapes (including kites and rhombuses) into rectangles and triangles, then add or subtract.

Shape	Area
Rectangle	$length \times width$ length × width
Triangle	$\frac{1}{2} \times b \times h$ 1/2 × b × h
Parallelogram	$b \times h$ b × h
Trapezium	$\frac{1}{2} (a + b) h$ 1/2(a + b)h, where $a$ a and $b$ b are the parallel sides

Worked example Measures · perimeter and area

Area of a trapezium

Question

A trapezium has parallel sides 7 cm and 13 cm. Its perpendicular height is 6 cm and each slanted side is 6.7 cm. Find its area.

1
Pick out the parallel sides and the perpendicular height: $a = 7$ a = 7, $b = 13$ b = 13, $h = 6$ h = 6
2
Add the parallel sides: $7 + 13 = 20$ 7 + 13 = 20
3
Multiply by the height and halve: $\frac{1}{2} \times 20 \times 6 = 60$ 1/2 × 20 × 6 = 60

Answer

60 cm²

⚠️ Watch out

using the slanted side 6.7 cm as the height gives 67 cm², which is wrong. The 6.7 cm is extra information.

Quick check

?A parallelogram has base 9 cm and slanted sides 5 cm. Its perpendicular height is 4 cm. What is its area?

Measures · perimeter and area

Circles

Circumference $C = π d = 2 π r$ C = π d = 2π r. Area $A = π r^{2}$ A = π r². Both are on the List of formulas.
Halve a diameter before using $π r^{2}$ π r²: squaring the diameter is the commonest slip.
Exact answers are left in terms of $π$ π: for $r = 3$ r = 3, $A = 9 π$ A = 9π cm².
A semicircle's perimeter is half the circumference plus the diameter.

Worked example Measures · perimeter and area

A track-shaped perimeter and area

Question

A shape is a 10 cm by 6 cm rectangle with a semicircle on each 6 cm end. Find its perimeter and its area, exactly and to 1 decimal place.

1
The two semicircles make one circle of diameter 6 cm: curved length $= π \times 6 = 6 π$ = π × 6 = 6π
2
Add the two straight sides: perimeter $= 20 + 6 π \approx 38.8$ = 20 + 6π ≈ 38.8 cm
3
The two semicircles make one circle of radius 3 cm: area $= π \times 3^{2} = 9 π$ = π × 3² = 9π
4
Add the rectangle: area $= 60 + 9 π \approx 88.3$ = 60 + 9π ≈ 88.3 cm²

Answer

perimeter

20 + 6 π \approx 38.8

cm; area

60 + 9 π \approx 88.3

cm²

⚠️ Watch out

the 6 cm sides are inside the shape, not on its edge, so they are not part of the perimeter.

Calculate

?Find the perimeter of a semicircle with diameter 12 cm. Give your answer to 1 decimal place.

Calculate

?A rectangular lawn is 12 m by 8 m. A semicircular flower bed with diameter 8 m is cut out of one end. Grass seed comes in boxes that each cover 15 m². How many boxes are needed to seed the lawn?

Measures · perimeter and area

Arcs and sectors

A sector is a "pizza slice" cut by two radii; the arc is its curved edge.
A sector with angle $θ$ θ is the fraction $\frac{θ}{360}$ θ/360 of the whole circle:

arc length = \frac{θ}{360} \times 2 π r sector area = \frac{θ}{360} \times π r^{2}

The perimeter of a sector is the arc plus two radii.
Core questions use angles that divide 360° (such as 90°, 60°, 40°); Extended uses any angle, including major sectors.

Calculate

?A sector has radius 9 cm and angle 40°. Find its area, to 1 decimal place.

cm²

Mission check

?Each lane on a track is 1.2 m wide, so lane 2's bends have a radius 1.2 m bigger than lane 1's. The straights are the same. About how much further is one lap in lane 2?

Mission 2

Roll a sheet into a tall tube or a wide one. Which holds more?

Make a guess: Roll a sheet into a tall tube or a wide one. Which holds more?

Measures · volume and surface area

Prisms and cylinders

A prism has the same cross-section all the way through. A cuboid has 6 faces, 12 edges and 8 vertices.
Volume of any prism:

V = area of cross-section \times length

Cuboid: $V = l \times w \times h$ V = l × w × h. Cylinder: the cross-section is a circle, so $V = π r^{2} h$ V = π r² h.
Units: cm³ or m³. 1 litre $= 1000$ = 1000 cm³, and 1 m³ $= 1000$ = 1000 litres.
Extended: recognise planes of symmetry and axes of symmetry of prisms, cylinders, pyramids and cones.

Worked example Measures · volume and surface area

Volume of a triangular prism

Question

A prism has a right-angled triangle as its cross-section, with shorter sides 6 cm and 8 cm. The prism is 15 cm long. Find its volume.

1
Area of the cross-section: $\frac{1}{2} \times 6 \times 8 = 24$ 1/2 × 6 × 8 = 24 cm²
2
Multiply by the length: $24 \times 15 = 360$ 24 × 15 = 360

Answer

360 cm³

⚠️ Watch out

forgetting to halve for the triangle gives

6 \times 8 \times 15 = 720

cm³, double the true volume.

Calculate

?A cylindrical water tank has radius 0.5 m and height 1.2 m. How many litres does it hold when full? Give your answer to the nearest litre.

litres

Measures · volume and surface area

Surface area, nets, plans and elevations

Surface area is the total area of the faces. A net shows them unfolded flat.
Cuboid: $S A = 2 (lw + l h + w h)$ SA = 2(lw + lh + wh).
A cylinder's net is two circles and a rectangle whose width is the circumference $2 π r$ 2π r. Curved surface $= 2 π r h$ = 2π r h; total $= 2 π r^{2} + 2 π r h$ = 2π r² + 2π r h.

Quick check

?A cuboid measures 5 cm by 4 cm by 3 cm. What is its total surface area?

Measures · volume and surface area

Pyramids, cones and spheres

Pyramid: $V = \frac{1}{3} \times base area \times h$ V = 1/3 × base area × h.
Cone: $V = \frac{1}{3} π r^{2} h$ V = 1/3π r² h, a third of the cylinder with the same base and height. Curved surface $= π r l$ = π r l, where the slant height $l$ l comes from Pythagoras: $l^{2} = r^{2} + h^{2}$ l² = r² + h².
Sphere: $V = \frac{4}{3} π r^{3}$ V = 4/3π r³ and $S A = 4 π r^{2}$ SA = 4π r². Half a sphere is a hemisphere.
Extended: a frustum is a cone with its top cut off: volume $=$ = large cone $-$ − small cone.
All these formulas are on the List of formulas.

Worked example Measures · volume and surface area

Volume and curved surface of a cone

Question

A cone has base radius 3 cm and perpendicular height 4 cm. Find its volume and its curved surface area, in terms of

π

and to 1 decimal place.

1
Volume: $\frac{1}{3} \times π \times 3^{2} \times 4 = 12 π \approx 37.7$ 1/3 × π × 3² × 4 = 12π ≈ 37.7 cm³
2
Compare: a cylinder with the same base and height holds $π \times 3^{2} \times 4 = 36 π$ π × 3² × 4 = 36π, three times as much
3
Slant height by Pythagoras: $l = 3^{2} + 4^{2} = 5$ l = √(3² + 4²) = 5 cm
4
Curved surface: $π r l = π \times 3 \times 5 = 15 π \approx 47.1$ π r l = π × 3 × 5 = 15π ≈ 47.1 cm²

Answer

volume

12 π \approx 37.7

cm³; curved surface area

15 π \approx 47.1

cm²

⚠️ Watch out

π r l

needs the slant height. Using the perpendicular height gives

π \times 3 \times 4 = 12 π

cm², which is too small.

Calculate

?A solid is a cylinder of radius 3 cm and height 10 cm with half a sphere (a hemisphere) of radius 3 cm on top. Find its volume, to 3 significant figures.

cm³

Mission check

?One 20 cm by 30 cm sheet is rolled into a tube two ways: tall (30 cm high) or short (20 cm high). Which holds more?

Recap

Big ideas

Area: rectangle $lw$ lw, triangle $\frac{1}{2} bh$ 1/2bh, parallelogram $bh$ bh, trapezium $\frac{1}{2} (a + b) h$ 1/2(a+b)h, always with the perpendicular height.
Circle: $C = 2 π r = π d$ C = 2π r = π d, $A = π r^{2}$ A = π r²; give exact answers in terms of $π$ π when asked; split compound shapes into parts.
Sector $= \frac{θ}{360}$ = θ/360 of the circle for both arc length and area.
Prism volume $=$ = cross-section area $\times$ × length; cylinder $V = π r^{2} h$ V = π r² h; 1 litre $= 1000$ = 1000 cm³.
Surface area is the sum of the faces: use a net; a cylinder's curved surface unrolls to a $2 π r$ 2π r by $h$ h rectangle.
Pyramid and cone volumes are a third of the matching prism or cylinder; sphere $V = \frac{4}{3} π r^{3}$ V = 4/3π r³, $S A = 4 π r^{2}$ SA = 4π r².

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Mission 1Why do runners in outer lanes start further ahead?
Mission 2Roll a sheet into a tall tube or a wide one. Which holds more?

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Specification points and sources for this lesson are listed at the bottom of the page.

Sources (2)

The specification points and sources behind this lesson.

1 Cambridge IGCSE Mathematics 0580 syllabus 2025–2027, C5.2, C5.3, C5.5, E5.2, E5.3, E5.5
2 Cambridge IGCSE Mathematics 0580 syllabus 2025–2027, C4.1, C4.2.3, C5.1, C5.4, C5.5, E4.1, E4.5.2, E5.4, E5.5