Mensuration
Pyramids, cones and spheres
In this note
- 5.4Find the volume of a pyramid, cone and sphere.
- 5.4Find the surface area of a cone (curved + base) and a sphere or hemisphere.
- 5.4Use Pythagoras for a cone's slant height, and work backwards to a radius.
Before you start: Prisms and cylinders and Pythagoras. Next: Compound solids.
Formula bank
Pyramids and cones
A pyramid or cone holds one third of the prism or cylinder with the same base and height. Square-based pyramid, base 6 cm, height 10 cm: \(\tfrac13\times36\times10=120\) cm³.
General methodHow to answer any cone question
- Find r, h and l. If one is missing, use Pythagoras: \(l^{2}=r^{2}+h^{2}\).
- Volume: \(\tfrac13\pi r^{2}h\) (uses the HEIGHT).
- Curved surface: \(\pi rl\) (uses the SLANT height). Total surface of a solid cone: \(\pi rl+\pi r^{2}\).
- Give the answer in terms of \(\pi\) or rounded as asked.
Example A cone has radius 5 cm and height 12 cm. Find its volume and total surface area.
Slant height first (Pythagoras).
- \(l=\sqrt{5^{2}+12^{2}}=13\)
- \(V=\tfrac13\pi(25)(12)=100\pi\approx314.16\)
- area \(=\pi(5)(13)+\pi(25)=90\pi\approx282.74\)
Answer 314.16 cm³ and 282.74 cm²
Try it yourself
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Answer 120
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Answer 56
Spheres and hemispheres
Radius 3 cm: \(V=\tfrac43\pi(27)=36\pi\approx113.10\) cm³ and surface \(4\pi(9)=36\pi\) cm². A hemisphere is half: \(V=\tfrac23\pi r^{3}\); a SOLID hemisphere's surface also includes the flat circle: \(2\pi r^{2}+\pi r^{2}=3\pi r^{2}\).
General methodHow to work backwards to a radius
- Put the given volume (or area) into the formula.
- Rearrange for \(r^{3}\) (volume) or \(r^{2}\) (area).
- Cube root (or square root).
Example A sphere has volume \(288\pi\) cm³. Find its radius.
\(\tfrac43\pi r^{3}=288\pi\).
- \(r^{3}=288\times\tfrac34=216\)
- \(r=\sqrt[3]{216}\)
Answer 6 cm
Try it yourself
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Answer 523.60
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Answer 108
Ways they test it
Type 1 Volume with a formula given
"[Volume of a cone \(=\tfrac13\pi r^{2}h\)] Calculate the volume." (2 marks)
Spot it a formula printed with the question.
- Substitute carefully; round at the end.
Type 2 Slant height needed
"A cone has radius 6 cm and slant height 10 cm. Find its volume." (3 marks)
Spot it slant height given, volume wanted (or the reverse).
- Pythagoras for the missing h or l.
- Then the formula.
For example \(h=\sqrt{100-36}=8\)
- \(V=\tfrac13\pi(36)(8)\)
Answer 301.59 cm³
Type 3 Find the radius
"A sphere has volume \(288\pi\). Find r." (2 marks)
Spot it volume or area given.
- Formula = value.
- Root.
Type 4 Hemisphere surface
"Find the total surface area of the solid hemisphere." (2 marks)
Spot it 'solid hemisphere'.
- Curved \(2\pi r^{2}\) + flat \(\pi r^{2}\).
Mistakes to avoid
- ✗ Cone volume with the slant height✓ use the vertical heightHeight must be perpendicular to the base.
- ✗ Curved surface with h✓ \(\pi rl\) uses the slant height
- ✗ Hemisphere surface = 2πr² for a solid✓ \(3\pi r^{2}\)Add the flat circle.
- ✗ Forgetting the ⅓✓ pyramid and cone volumes have \(\tfrac13\)
Summary
- Pyramid/cone volume = \(\tfrac13\) × base × height.
- Cone: \(l^{2}=r^{2}+h^{2}\); curved surface \(\pi rl\).
- Sphere: \(\tfrac43\pi r^{3}\), \(4\pi r^{2}\); solid hemisphere surface \(3\pi r^{2}\).
If you see …
- a cone with h and l mixedPythagoras first.
- a volume givenformula = value, then cube root.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 36
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Answer 13
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Answer 65
Medium
Guide me
What is it asking? a volume
What do we know? \(\tfrac13\pi r^{2}h\)
Plan \(\tfrac13\pi(25)(12)\)
Show working
- \(100\pi\)
Answer 314.16
Guide me
What is it asking? a volume
What do we know? \(\tfrac23\pi r^{3}\)
Plan \(\tfrac23\pi(216)\)
Show working
- \(144\pi\)
Answer 452.39
Guide me
What is it asking? a radius
What do we know? \(\tfrac43r^{3}=288\)
Plan \(r^{3}=216\)
Show working
- 6
Answer 6
Hard
Guide me
What is it asking? a volume
What do we know? height by Pythagoras: 8
Plan \(\tfrac13\pi(36)(8)\)
Show working
- \(h=8\)
- \(96\pi=301.59\)
Answer 301.59
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| h = 8 | M1 | ||
| \(\tfrac13\pi\times36\times8\) | M1 | ||
| 301.59 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? an area
What do we know? curved \(2\pi r^{2}=72\pi\)
Plan + flat \(36\pi\)
Show working
- \(108\pi=339.29\)
Answer 339.29
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(2\pi(36)+\pi(36)\) | M1 | ||
| 339.29 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? an area
What do we know? \(4\pi r^{2}\)
Plan \(100\pi\)
Show working
- 314.16
Answer 314.16
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 314.16 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can find volumes of pyramids and cones and surface areas of cones.
- I can find volumes and surface areas of spheres and hemispheres.
- I can use Pythagoras for slant heights and work back to a radius.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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