A 3D solid with two identical parallel ends, called cross-sections, joined by straight rectangular sides. The cross-section can be any 2D shape — rectangle, triangle, trapezium, or circle (for a cylinder).

Year 11 General Shape And Measurement

Volume Of A Prism

Q: What is the universal formula for the volume of a prism?

V equals the cross-section area times the length. This works for any prism — rectangular, triangular, trapezoidal, or cylindrical.

Q: What is the formula for the volume of a rectangular prism?

Length times width times height. It's the same as cross-section area (length times width) times the height of the prism.

Q: What is the formula for the volume of a cylinder?

V equals pi r squared times h. The cross-section is a circle with area pi r squared, and h is the height of the cylinder.

Q: How do you convert cubic centimetres to litres?

Divide by 1,000. There are 1,000 cubic centimetres in 1 litre. So 1,570 cubic centimetres equals 1.570 litres.

Q: How do you find a missing dimension if you know the volume?

Use the volume formula and solve for the unknown. For a rectangular prism, divide volume by the two known dimensions to get the third.

13 practice questions 2 video lessons Theory + worked examples

Theory

A prism is a 3D solid with two identical parallel ends (the cross-sections) joined by straight sides. A cylinder is just a prism with a circular cross-section. The universal formula is $V = A_{cross-section} \times ℓ$ . Capacity is volume in litres: $1,000$ cm³ $=$ $1$ L.

A prism is a 3D solid with two identical, parallel cross-sections (the "ends") joined by straight rectangular sides. A cylinder is the special case where the cross-section is a circle.

The universal volume formula for any prism is $V = A_{cross-section} \times ℓ$ , where $ℓ$ is the length (the perpendicular distance between the two ends). This single formula covers rectangular, triangular, trapezoidal, and cylindrical prisms — just plug in the appropriate area for the cross-section.

Capacity is volume measured in litres or millilitres for fluids. Use $1,000$ cm³ $= 1$ L and $1$ m³ $= 1,000$ L $= 1$ kL. Compound solids are split into prisms whose volumes you compute separately, then add (or subtract for hollow parts).

Watch the two heights. A triangular prism has a triangle height (inside the cross-section) AND a prism length. They're different — label them clearly.

Same formula for every prism

Any prism: V = (this area) × length

The universal prism formula:

V = A_{cross-section} \times ℓ

V = A_{cross} \times ℓ

Cross-section areas for common prisms:

Prism	Volume
Rectangular (cuboid)	$V = ℓ \cdot w \cdot h$
Triangular	$V = \frac{1}{2} \cdot b \cdot h \cdot ℓ$
Trapezoidal	$V = \frac{1}{2} (a + b) \cdot h \cdot ℓ$
Cylinder	$V = π r^{2} \cdot h$

V = π r^{2} h

Capacity conversions: 1,000 cm³ =1 L; 1 m³ =1,000 L =1 kL.

How to find the volume of any prism

Identify the cross-section — the shape of the two parallel ends.
Find its area using the appropriate 2D formula (rectangle, triangle, trapezium, circle).
Multiply by the length of the prism (perpendicular distance between the two ends).
Convert if needed — cm³ to litres by dividing by $1,000$ ; m³ to litres by multiplying by $1,000$ .
For compound solids, split into prisms, find each volume, and add (or subtract hollow parts).

Example 1 — Rectangular prism

A box is

15

cm long,

8

cm wide, and

6

cm tall. Find its volume.

Solution

Use $V = ℓ \cdot w \cdot h$ .

$V$	$=$	$15 \times 8 \times 6$
$V$	$=$	$720$ cm³

V = 720

The volume is $720$ cm³.

Example 2 — Triangular prism

A triangular prism has a cross-section with base

10

cm and height

4

cm. The prism is

12

cm long. Find its volume.

Solution

Cross-section area first, then multiply by length.

$A_{△}$	$=$	$\frac{1}{2} \times 10 \times 4 = 20$ cm²
$V$	$=$	$20 \times 12$
$V$	$=$	$240$ cm³

V = 240

The volume is $240$ cm³.

Example 3 — Cylinder, capacity

A cylindrical tank has radius

0.5

m and height

2

m. Find its capacity in litres (2 dp).

Solution

Use $V = π r^{2} h$ , then convert m³ to litres.

$V$	$=$	$π \times {0.5}^{2} \times 2$
$V$	$=$	$0.5 π$
$V$	$\approx$	$1.5708$ m³
$cap.$	$=$	$1.5708 \times 1,000$
$cap.$	$\approx$	$1,570.80$ L

capacity \approx 1570.80

The capacity is about $1{,}570.80$ L.

Example 4 — Reverse: find a dimension

A rectangular tank is

4

m long,

2

m wide, and holds a volume of

12

m³. Find its height.

Solution

Substitute into $V = ℓ w h$ and solve for $h$ .

$V$	$=$	$ℓ \cdot w \cdot h$
$12$	$=$	$4 \times 2 \times h$
$12$	$=$	$8 h$
$h$	$=$	$\frac{12}{8} = 1.5$ m

h = 1.5

The height is $1.5$ m.

Common pitfalls

Confusing the two heights in a triangular prism. The triangle's perpendicular height (inside the cross-section) and the prism's length are different. Label them as you go.

Using diameter instead of radius for a cylinder.

V = π r^{2} h

. If you're given the diameter, halve it first.

Unit conversion for capacity.

1,000

cm³

= 1

L, and

1

m³

= 1,000

L. Mixing the two factors gives an answer off by a thousand.

Forgetting the cross-section must be the parallel ends. The cross-section is whichever shape appears identically at both ends — not always the "bottom".

Frequently asked questions

What is a prism?

A 3D solid with two identical parallel ends (cross-sections) joined by straight sides. The cross-section can be any 2D shape.

What is the universal formula for the volume of a prism?

$V = A_{cross-section} \times ℓ$ . Works for every prism — rectangular, triangular, trapezoidal, or cylindrical.

What is the formula for the volume of a rectangular prism?

$V = ℓ \cdot w \cdot h$ — length times width times height.

What is the formula for the volume of a cylinder?

$V = π r^{2} h$ . The circular cross-section has area $π r^{2}$ , times the height $h$ .

How do you convert cubic centimetres to litres?

Divide by $1,000$ . $1,000$ cm³ $= 1$ L. So $1,570$ cm³ $= 1.570$ L.

How do you find a missing dimension if you know the volume?

Substitute the known dimensions and volume into the formula, then solve for the unknown by division.

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