Volume of Prisms and Surface Area

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Volume tells you how much space a solid takes up (how many 1 cm cubes would fill it). Surface area tells you the total area of all the faces on the outside (how much paper you would need to wrap it with no overlaps). They are different ideas with different units: volume is measured in cubic units (cm³, m³) and surface area in square units (cm², m²). On this page you will build the volume formula for a triangular prism from the cuboid, and then learn a reliable method for the surface area of any cube, cuboid, triangular prism or pyramid.

1. Key words

2. Volume of a cuboid — the starting point

A cuboid that is 5 cm long, 3 cm wide and 2 cm high holds 5 × 3 = 15 cubes in its bottom layer, and there are 2 layers, so its volume is 15 × 2 = 30 cm³. In general:

• Volume of a cuboid = length × width × height
• The bottom layer (5 × 3 = 15 cm²) is the area of the cross-section, and the height (2 cm) is how far that cross-section runs. So we can also say: volume of a cuboid = area of cross-section × length.
• Volume of a cube with edge a = a × a × a = a³.

3. Deriving the volume of a triangular prism

Take a cuboid with base b, height h and length l. Cut it straight through from one edge to the opposite edge (along the diagonals of the two end faces). You get two identical right-angled triangular prisms, so each one has exactly half the volume of the cuboid.

1. Volume of the cuboid = b × h × l.
2. Volume of one triangular prism = ½ × b × h × l.
3. But ½ × b × h is the area of the triangle at the end (the cross-section).
4. So: volume of a triangular prism = area of the triangular cross-section × length = ½ × b × h × l.

This also works when the triangle is not right-angled: any triangle with base b and perpendicular height h has area ½ × b × h (you met this when you found the area of a triangle as half of a rectangle). The prism is just that triangle 'pushed along' a length l. The rule volume = area of cross-section × length works for every prism: cube, cuboid, triangular prism, and later prisms with other cross-sections.

4. Surface area — add up every face

To find a surface area, imagine unfolding the solid into its net. The surface area is the total area of the net. Work face by face and list the faces so you do not miss one.

A neat trick for triangular prisms: the three rectangles all have the same length l, and their widths are the three sides of the triangle. So the three rectangles together = perimeter of the triangle × length.

For a pyramid, the triangular faces slope. The height you need for each triangle is its slant height (from the apex down the middle of the face to the base edge), not the vertical height of the pyramid.

Method for any surface-area question

1. Name the solid and count its faces (cube 6, cuboid 6, triangular prism 5, square-based pyramid 5).
2. Sketch the net or list the faces in groups of equal faces.
3. Find the area of each different face (rectangle = l × w, triangle = ½ × b × h).
4. Multiply by how many of each face there are, and add.
5. Check you have the right number of faces, and write the answer in square units.

5. Worked examples

Worked example 1 — The triangular prism in the diagram above has a triangle with base 6 cm and height 4 cm, and length 10 cm. Work out its volume. [2]

1. Area of the triangle = ½ × 6 × 4 = 12 cm².
2. Volume = area of cross-section × length = 12 × 10 = 120 cm³.

Common mistake: 6 × 4 × 10 = 240 cm³ — this forgets the ½ and gives the volume of the whole cuboid.

Worked example 2 — A triangular prism has volume 180 cm³. Its cross-section is a triangle with base 8 cm and perpendicular height 5 cm. Find the length of the prism. Show your working. [2]

1. Area of the triangle = ½ × 8 × 5 = 20 cm².
2. Volume = area × length, so 180 = 20 × length.
3. Length = 180 ÷ 20 = 9 cm.
4. Check: 20 × 9 = 180 ✓

Worked example 3 — Find the surface area of a cuboid 5 cm long, 3 cm wide and 2 cm high. [2]

Common mistake: adding 15 + 10 + 6 = 31 cm² and forgetting that each face has an equal opposite face. Compare: its volume is 5 × 3 × 2 = 30 cm³ — a different number and a different unit.

Worked example 4 — The net above belongs to a triangular prism with a right-angled triangle of sides 3 cm, 4 cm and 5 cm and length 10 cm. Work out (a) its surface area [2] and (b) its volume. [1]

1. (a) Two triangles: 2 × (½ × 3 × 4) = 2 × 6 = 12 cm². (The right angle is between the 3 cm and 4 cm sides, so these are the base and height.)
2. Three rectangles: 3 × 10 + 4 × 10 + 5 × 10 = 30 + 40 + 50 = 120 cm² (or perimeter × length = 12 × 10 = 120 cm²).
3. Surface area = 12 + 120 = 132 cm².
4. (b) Volume = 6 × 10 = 60 cm³.

Common mistake: using the 5 cm side as the height of the triangle. The 5 cm side is the sloping side (hypotenuse); the base and perpendicular height are 3 cm and 4 cm.

Worked example 5 — A square-based pyramid has a base of side 6 cm. Each triangular face has a slant height of 5 cm. Work out the total surface area. [2]

1. Base: 6 × 6 = 36 cm².
2. One triangle: ½ × 6 × 5 = 15 cm². Four triangles: 4 × 15 = 60 cm².
3. Surface area = 36 + 60 = 96 cm².

Worked example 6 (thinking mathematically) — A cube has a surface area of 150 cm². Work out its volume. [2]

1. A cube has 6 equal square faces, so one face = 150 ÷ 6 = 25 cm².
2. Each edge = √25 = 5 cm.
3. Volume = 5 × 5 × 5 = 125 cm³.

Common mistake: dividing 150 by 6 and then cubing 25. You must find the edge first (the square root of the face area).

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3