Geometry & Measures
Grade 4-5Volume & Surface Area of Prisms
A prism is any 3D shape with a constant cross-section along its length, which means the same volume formula works for cuboids, triangular prisms and trapezium-cross-section prisms alike. This lesson covers the volume and surface area of cuboids, finding the volume of a prism from its cross-sectional area, and working backwards from a given volume to find a missing length.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Volume and surface area of a cuboid
The volume of a cuboid is length × width × height. Its surface area is the total area of all six rectangular faces, found by adding the areas of the three different pairs of faces and doubling the result, since each face has an identical opposite face.
A cuboid measuring 8 cm by 5 cm by 4 cm has a volume of 8 × 5 × 4 = 160 cm³. Its three different face areas are 8×5=40, 8×4=32 and 5×4=20, so its surface area is 2 × (40 + 32 + 20) = 184 cm².
Volume of a prism from its cross-section
Every prism has the same volume formula: cross-sectional area × length. The cross-section can be any 2D shape, most commonly a triangle, trapezium or L-shape, and its area is found using the appropriate area formula before multiplying by the length of the prism.
A triangular prism has a cross-section with base 6 cm and height 4 cm, and a length of 10 cm. Its cross-sectional area is ½ × 6 × 4 = 12 cm², so its volume is 12 × 10 = 120 cm³.
Surface area of a triangular prism
The surface area of a triangular prism is the area of its two triangular ends, plus the area of its three rectangular side faces, which together equal the triangle perimeter multiplied by the prism length.
A triangular prism has a right-angled cross-section with legs 3 cm and 4 cm and hypotenuse 5 cm, and a length of 10 cm. The two triangular ends have a combined area of 2 × (½ × 3 × 4) = 12 cm², and the three rectangular faces have a combined area of (3 + 4 + 5) × 10 = 120 cm², giving a total surface area of 132 cm².
Working backwards from a given volume
If the volume and some of the dimensions of a prism are known, divide the volume by the known cross-sectional area or by the known length and width to find the missing length.
A cuboid has a volume of 150 cm³, a length of 10 cm and a width of 5 cm. Since 10 × 5 = 50, the height is 150 ÷ 50 = 3 cm.
Worked Examples
Three exam-style questions, fully solved.
Find the volume of a cuboid measuring 8 cm by 5 cm by 4 cm.
Easy- 1.Multiply the three dimensions together: 8 × 5 × 4
Answer: 160 cm³
A cuboid has a volume of 150 cm³. Its length is 10 cm and its width is 5 cm. Find its height.
Medium- 1.Find the area of the base: 10 × 5 = 50 cm²
- 2.Divide the volume by the base area to find the height: 150 ÷ 50
Answer: h = 3 cm
The triangular cross-section of a prism is right-angled, with legs 3 cm and 4 cm and hypotenuse 5 cm. The prism has a length of 10 cm. Find the surface area of the prism.
Hard- 1.Find the area of one triangular end: ½ × 3 × 4 = 6 cm²
- 2.Find the perimeter of the triangular end: 3 + 4 + 5 = 12 cm
- 3.Find the total surface area: (2 × 6) + (12 × 10)
Answer: 132 cm²
Avoid These
The most common mistakes students make.
Forgetting to double the sum of the three face areas when finding the surface area of a cuboid, since each face has an identical opposite face.
Using the cross-sectional area formula for a rectangle instead of a triangle or trapezium when the prism has a triangular or trapezium cross-section.
Confusing the length of the prism with a dimension of the cross-section, and multiplying the wrong two measurements together.
When finding the surface area of a triangular prism, forgetting to include both triangular end faces, or using the triangle area instead of its perimeter for the rectangular side faces.
When working backwards from a volume, dividing by only one dimension instead of the full base area or cross-sectional area.
FAQ
Questions parents and students ask.
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