Mensuration — 3D Solids

Volume, curved surface and total surface of the cuboid, cube, cylinder, hollow cylinder, cone, frustum, sphere and hemisphere, with the prism and pyramid in their general form; the slant height relation for a cone; melting and recasting one solid into another; combined solids such as a cone on a cylinder; water flow through a pipe into a tank; effect on volume when a dimension changes; capacity in litres — with fully worked examples.

Mensuration — 3D Solids

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📑 Contents (5 sections)

Last reviewed 22 Sept 2026 · 5 min read

The standard solids

FormulaVolumes and surfaces

Solid	Volume	Curved / lateral surface	Total surface
Cuboid	$l bh$	$2 h (l + b)$	$2 (l b + bh + h l)$
Cube	$a^{3}$	$4 a^{2}$	$6 a^{2}$
Cylinder	$π r^{2} h$	$2 π r h$	$2 π r (r + h)$
Cone	$\frac{1}{3} π r^{2} h$	$π r l$	$π r (r + l)$
Sphere	$\frac{4}{3} π r^{3}$	—	$4 π r^{2}$
Hemisphere	$\frac{2}{3} π r^{3}$	$2 π r^{2}$	$3 π r^{2}$

Diagonal of a cuboid $= l^{2} + b^{2} + h^{2}$ ; of a cube $= a 3$ .

FormulaCone, frustum, prism and pyramid

Cone slant height: $l = r^{2} + h^{2}$
Frustum of a cone (radii $R$ and $r$ , height $h$ ): $V = \frac{π h}{3} (R^{2} + R r + r^{2})$ , slant $l = h^{2} + (R - r)^{2}$ , curved surface $= π l (R + r)$
Hollow cylinder (outer $R$ , inner $r$ ): $V = π h (R^{2} - r^{2})$
Prism: $V =$ base area $\times$ height; lateral surface $=$ base perimeter $\times$ height
Pyramid: $V = \frac{1}{3} \times$ base area $\times$ height; lateral surface $= \frac{1}{2} \times$ base perimeter $\times$ slant height

Capacity and units

1 m^{3} = 1000 litres, 1 litre = 1000 cm^{3}

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