Learning to calculate the volume and surface area of 3D...
Maths246 views·Updated 20 Aug 2026·4 pagesSurface Area and Volume Formulas
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atty Ali@attyali_jgch
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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 πr³. For a ball with radius 7cm, you'd get × 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 πr²h, where h is the perpendicular height (straight up from base to tip). For a cone with radius 5cm and height 8cm: × 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 20 Aug 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...
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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
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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 πr³. For a ball with radius 7cm, you'd get × 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
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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 4Sign up to see the content. It's free!
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By signing up you accept Terms of Service and Privacy Policy
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 πr²h, where h is the perpendicular height (straight up from base to tip). For a cone with radius 5cm and height 8cm: × 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!
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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1080,2396,325
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This educational resource provides a comprehensive overview of calculating the surface area and volume of various 3D shapes, including cuboids and cylinders. Ideal for Year 8 students, it covers essential formulas and step-by-step examples to enhance understanding of geometric concepts.
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Stefan SiOS user
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