📐 Grade 8 · Lesson 14 of 15
Volume Workshop: Cylinders, Cones, and Spheres
Three curved solids, three formulas — and a surprisingly tidy relationship linking them all.
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The big idea: Volume measures how much space a solid fills. A cylinder's volume is just the area of its circular base stacked up through its height. A cone with the same base and height holds exactly one-third of that — no more, no less. And a sphere's volume follows its own formula built from the same radius. Once you know the radius (and height, where it applies), each formula turns a shape into a number.
Volume: how much space is inside
Volume measures how much three-dimensional space a solid takes up — how much water it could hold, or how much material it's made of. For shapes built from flat faces, like a rectangular box, volume is just length × width × height. Curved solids — cylinders, cones, and spheres — use a related idea, built around the circle.
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A cylinder is a stack of circles
A cylinder is what you get by stacking identical circles straight up. Its volume is the area of one circular base (πr²) multiplied by how high the stack goes.
\[ V_{\text{cylinder}} = \pi r^2 h \]
r is the radius of the circular base; h is the height of the cylinder.
🎮 Volume Workshop LIVE
Switch between cone, cylinder and sphere and see how each volume formula relates.
A cone is one-third of its matching cylinder
A cone with the same circular base and the same height as a cylinder holds exactly one-third as much space. This isn't a coincidence to memorize blindly — it's a genuine geometric fact: fill a cone with sand or water and pour it into a cylinder of the same base and height, and it takes exactly three cones to fill the cylinder.
\[ V_{\text{cone}} = \dfrac{1}{3}\pi r^2 h \]
Same πr²h as the cylinder, scaled down by one-third.
📝 Worked example: Find the volume of a cylinder with radius 3 and height 10 (leave your answer in terms of π, then estimate using π ≈ 3.14).
- Apply the formula: V = πr²h = π(3²)(10).
- 3² = 9, so V = π(9)(10) = 90π.
- Using π ≈ 3.14: V ≈ 90 × 3.14 = 282.6.
✓ The cylinder's volume is <strong>90π</strong>, about <strong>282.6 cubic units</strong>.
📝 Worked example: Find the volume of a cone with the same radius (3) and height (10) as the cylinder above.
- Apply the formula: V = (1/3)πr²h = (1/3)π(9)(10) = (1/3)(90π).
- (1/3) of 90π is 30π.
- Using π ≈ 3.14: V ≈ 30 × 3.14 = 94.2.
✓ The cone's volume is <strong>30π</strong>, about <strong>94.2 cubic units</strong> — exactly one-third of the cylinder's 90π, as expected.
A sphere: its own formula, same radius
A sphere (a perfectly round ball) doesn't have a height to plug in — it only needs its radius. Its volume formula uses the radius cubed, scaled by a factor of four-thirds.
\[ V_{\text{sphere}} = \dfrac{4}{3}\pi r^3 \]
r is the sphere's radius, measured from the centre to the surface.
📝 Worked example: Find the volume of a sphere with radius 6.
- Apply the formula: V = (4/3)πr³ = (4/3)π(6³).
- 6³ = 216, so V = (4/3)π(216).
- (4/3) of 216 is 288, so V = 288π ≈ 288 × 3.14 = 904.3.
✓ The sphere's volume is <strong>288π</strong>, about <strong>904.3 cubic units</strong>.
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Match the right formula to the right shape — and don't drop the fraction
The three formulas look similar but are easy to mix up: cylinders use just πr²h, cones need the extra factor of 1/3, and spheres use r³ (not r²) with a factor of 4/3. The most common mistake is forgetting the 1/3 on a cone or the 4/3 on a sphere — always double-check you copied the whole formula, not just πr²h or πr³ on their own.
Check your understanding
1. What is the volume of a cylinder with radius 4 and height 5? (Leave in terms of π.)
V = πr²h = π(4²)(5) = π(16)(5) = 80π.
2. A cone and a cylinder share the same radius and the same height. How do their volumes compare?
A cone always holds exactly one-third the volume of a cylinder with the same base and height.
3. What is the volume of a sphere with radius 3? (Leave in terms of π.)
V = (4/3)πr³ = (4/3)π(27) = 36π, since 3³ = 27 and (4/3)(27) = 36.
4. Which formula correctly gives the volume of a cylinder?
A cylinder's volume is the circular base's area (πr²) times its height (h).
5. When finding the volume of a cone, what is the most common mistake to avoid?
It's easy to compute πr²h (the cylinder's volume) and forget the final step of multiplying by 1/3 to get the cone's volume.
✅ Key takeaways
- Volume measures how much three-dimensional space a solid fills.
- A cylinder's volume is its circular base's area times its height: V = πr²h.
- A cone with the same base and height as a cylinder holds exactly one-third the volume: V = (1/3)πr²h.
- A sphere's volume depends only on its radius, cubed: V = (4/3)πr³.
- The three formulas are easy to confuse — always check you've used the right exponent and the right fraction for the shape.