Learning to calculate the volume and surface area of 3D... Show more
Maths246 views·Updated May 29, 2026·4 pagesSurface Area and Volume Formulas
A
atty Ali@attyali_jgch
1 / 4
1
of 4
Cylinders
Think of a cylinder as a tin can - it's got a circular top and bottom with curved sides connecting them. The volume formula is π r² h, where r is the radius and h is the height.
For surface area, you need to add up all the surfaces: the curved side plus the two circular ends. The formula becomes 2πrh + 2πr². The first part (2πrh) gives you the curved surface, whilst the second part (2πr²) covers both circular ends.
Let's say you've got a cylinder with radius 5cm and height 15cm. For surface area: 2 × π × 5 × 15 + 2 × π × 5² = 628cm². For volume of a water tank with radius 25cm and height 120cm: π × 25² × 120 = 235,500cm³.
Quick tip: Remember that volume always uses cubic units (cm³) whilst surface area uses square units (cm²).
2
of 4
Spheres
A sphere is perfectly round like a football or tennis ball. Unlike other 3D shapes, spheres only need the radius to calculate everything - no height or width measurements required.
The volume formula is (4/3)πr³. For a ball with radius 7cm, you'd get (4/3) × π × 7³ = 436.75cm³. That's quite a lot of space inside!
Surface area uses 4πr². If your sphere has a radius of 10cm, the calculation becomes 4π × 10² = 502.4cm². This formula is much simpler than cylinders because there's only one curved surface to worry about.
Remember: Spheres are the only 3D shape where you don't need to think about separate faces - it's all one smooth surface.
3
of 4
Pyramids
Pyramids have a flat base (usually square) with triangular faces that meet at a point at the top. Think of the Egyptian pyramids, but smaller and with precise measurements!
The volume formula is (length × width × height) ÷ 3. For a pyramid that's 8cm by 6cm at the base and 9cm tall, you'd calculate: (8 × 6 × 9) ÷ 3 = 144cm³. Notice how it's one-third of what a rectangular box would be.
Surface area means adding up all the faces. You need the area of the square base plus all four triangular sides. For a pyramid with base 5cm × 5cm and triangular faces of 20cm² each: base area = 25cm², four triangular faces = 80cm², total = 105cm².
Pro tip: Always remember to divide by 3 for pyramid volume - it's the most common mistake students make!
4
of 4
Cones
A cone looks like an ice cream cone or traffic cone - it has a circular base that tapers to a point. You'll need to know the radius of the base and either the height or slant height.
The volume formula is (1/3)πr²h, where h is the perpendicular height (straight up from base to tip). For a cone with radius 5cm and height 8cm: (1/3) × π × 5² × 8 = 209.3cm³.
Surface area combines the circular base (πr²) with the curved surface (πrl, where l is the slant height). With radius 14cm and slant height 8cm, you'd calculate: π × 14² + π × 14 × 8 = 508cm².
Key point: Make sure you know whether you're given the perpendicular height or slant height - they're different measurements!
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Maths246 views·Updated May 29, 2026·4 pagesSurface Area and Volume Formulas
A
atty Ali@attyali_jgch
Learning to calculate the volume and surface area of 3D shapes is essential for GCSE maths and real-world applications. These formulas help you work out everything from the amount of paint needed to cover a cylindrical tank to the volume... Show more
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Cylinders
Think of a cylinder as a tin can - it's got a circular top and bottom with curved sides connecting them. The volume formula is π r² h, where r is the radius and h is the height.
For surface area, you need to add up all the surfaces: the curved side plus the two circular ends. The formula becomes 2πrh + 2πr². The first part (2πrh) gives you the curved surface, whilst the second part (2πr²) covers both circular ends.
Let's say you've got a cylinder with radius 5cm and height 15cm. For surface area: 2 × π × 5 × 15 + 2 × π × 5² = 628cm². For volume of a water tank with radius 25cm and height 120cm: π × 25² × 120 = 235,500cm³.
Quick tip: Remember that volume always uses cubic units (cm³) whilst surface area uses square units (cm²).
2
of 4Sign up to see the content. It's free!
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Spheres
A sphere is perfectly round like a football or tennis ball. Unlike other 3D shapes, spheres only need the radius to calculate everything - no height or width measurements required.
The volume formula is (4/3)πr³. For a ball with radius 7cm, you'd get (4/3) × π × 7³ = 436.75cm³. That's quite a lot of space inside!
Surface area uses 4πr². If your sphere has a radius of 10cm, the calculation becomes 4π × 10² = 502.4cm². This formula is much simpler than cylinders because there's only one curved surface to worry about.
Remember: Spheres are the only 3D shape where you don't need to think about separate faces - it's all one smooth surface.
3
of 4Sign up to see the content. It's free!
- Access to all documents
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- Join milions of students
Pyramids
Pyramids have a flat base (usually square) with triangular faces that meet at a point at the top. Think of the Egyptian pyramids, but smaller and with precise measurements!
The volume formula is (length × width × height) ÷ 3. For a pyramid that's 8cm by 6cm at the base and 9cm tall, you'd calculate: (8 × 6 × 9) ÷ 3 = 144cm³. Notice how it's one-third of what a rectangular box would be.
Surface area means adding up all the faces. You need the area of the square base plus all four triangular sides. For a pyramid with base 5cm × 5cm and triangular faces of 20cm² each: base area = 25cm², four triangular faces = 80cm², total = 105cm².
Pro tip: Always remember to divide by 3 for pyramid volume - it's the most common mistake students make!
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Cones
A cone looks like an ice cream cone or traffic cone - it has a circular base that tapers to a point. You'll need to know the radius of the base and either the height or slant height.
The volume formula is (1/3)πr²h, where h is the perpendicular height (straight up from base to tip). For a cone with radius 5cm and height 8cm: (1/3) × π × 5² × 8 = 209.3cm³.
Surface area combines the circular base (πr²) with the curved surface (πrl, where l is the slant height). With radius 14cm and slant height 8cm, you'd calculate: π × 14² + π × 14 × 8 = 508cm².
Key point: Make sure you know whether you're given the perpendicular height or slant height - they're different measurements!
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Students love us — and so will you.
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Stefan SiOS user
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