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Volumes of Prism, Cylinder, Pyramid, Cone, Sphere and Spherical Cap

Volume is the space inside a solid, counted in cubic units. Prism and cylinder: base area × height. Pyramid and cone: one third of that. Sphere: (4/3)πr³, which is 2/3 of the cylinder around it. A spherical cap of height k has volume πk²(3r − k)/3. Cavalieri's idea explains all of them: equal slice areas at every height give equal volumes.

🎬 Step-by-step story

  1. A prism is a stack of thin slices. Every slice has the same area. Volume = base area × height.
  2. A cylinder is a stack of circles. Each circle has area πr². So volume = πr²h.
  3. A pyramid fits inside a prism with the same base and height. Its slices get smaller going up. It fills exactly one third of the prism.
  4. A cone fills exactly one third of the cylinder around it. Same idea, round base.
  5. A sphere of radius r fits inside a cylinder of height 2r. It fills two thirds of it. That gives V = (4/3)πr³.
  6. Your turn. Pick a shape, change r, and slide the orange slice up and down. Read the slice area and the volume. Try the spherical cap.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why is volume in cubic units?

Volume counts how many unit cubes fit in. Each slice has an area (length × length) and a thickness (a length), so three lengths multiply: cm × cm × cm = cm³.

Does a slanted prism have the same volume as a straight one?

Yes, if the base and the vertical height are the same. Every slice is still the same size, so Cavalieri's idea says the volume is the same. Use the vertical height, not the slanted edge.

Why exactly one third for a pyramid?

The pyramid slices shrink as you go up, from the full base to a point. Adding them all gives exactly one third of the prism's equal slices. Move the slice and watch its area fall.

Which height goes in the cone formula, slant or vertical?

The vertical height from apex straight down to the base. If you know the slant height l, use h = √(l² − r²).

Why does the ball formula have r³ and a 4/3?

Volume always needs three lengths, so r³. The 4/3 comes from the ball being 2/3 of the cylinder 2πr³ around it.

How big is a spherical cap?

It depends on the cap height k and the ball radius r: V = πk²(3r − k)/3. Choose Spherical cap and slide k to see k = r give a hemisphere.

What is volume? Slicing and Cavalieri's idea

Volume is the amount of space inside a solid. We count it in cubic units: cm³, m³, litres (1 litre = 1000 cm³).

Cut a solid into very thin slices, like slices of bread. If you know the area of each slice and its height, you can add them up. The Italian mathematician Cavalieri noticed a simple fact: if two solids have the same height, and at every height their slices have the same area, then their volumes are equal. That is why a leaning stack of coins has the same volume as a straight stack.

Prism and cylinder: base area × height

A prism has two equal parallel faces (bases) joined by rectangles. Every slice parallel to the base is a copy of the base, so V = B × h, where B is the base area and h is the perpendicular height. For a box, B = l × b.

A cylinder has circular bases, so B = πr² and V = πr²h. Even a slanted (oblique) prism or cylinder has V = B × h, but h must be the straight up-and-down height, not the slanted edge.

Pyramid and cone: why one third?

A pyramid has one base and triangular faces that meet at a point (apex). A cone is the round version. Take a pyramid and a prism with the same base and the same height. At height t from the base the pyramid slice is a smaller copy of the base, scaled by (1 − t/h). Its area is B(1 − t/h)². Adding all the slices gives exactly ⅓ B h. (A cube can be cut into 3 equal pyramids that meet at one corner: this shows the one third without any calculus.)

So pyramid: V = ⅓ × B × h and cone: V = ⅓πr²h. Here h is the vertical height, not the slant height l. If you are given l, first find h = √(l² − r²).

Ball (sphere): V = (4/3)πr³

A sphere is the surface; the ball is the solid inside it. Put a ball of radius r inside a cylinder of radius r and height 2r. Also put two cones inside the cylinder, tips meeting at the centre. At height y from the centre, the cylinder slice is a circle of area πr². The ball slice has area π(r² − y²). The two-cone slice has area πy². These two add up to πr², so at every height ball + cones = cylinder. By Cavalieri, the volumes add the same way: cylinder = 2πr³, the two cones together = 2πr³/3, so the ball = 2πr³ − 2πr³/3 = (4/3)πr³. It is exactly 2/3 of the cylinder. A hemisphere is half of that: (2/3)πr³.

Ball segment (spherical cap)

Cut a ball with a flat plane. The smaller piece is a spherical cap (also called a ball segment with one base). If the cap has height k and the ball has radius r, then V = πk²(3r − k)/3. Check: when k = 2r the cap is the whole ball: π(4r²)(r)/3 = (4/3)πr³. When k = r it is a hemisphere: πr²(2r)/3 = (2/3)πr³.

The flat face of the cap is a circle of radius a where a² = k(2r − k). A segment between two parallel planes is a difference of two caps. In the 3D, choose Spherical cap and slide the cap height k.

Which formula to use: a quick table

Always check that all lengths use the same unit before you multiply. Then write the answer in cubic units.

Try it: the rice test

Take a paper cone and a paper cylinder of the same width and the same height (stick-paper or card). Fill the cone with rice and pour it into the cylinder. Do it three times: the cylinder is full. Now predict first: how many cones of rice fill it? Then check. You have just measured the one third yourself.

Key formulas and definitions

Worked examples

1. A square prism has base edge 4 cm and height 9 cm. Find its volume.

B = 4 × 4 = 16 cm². V = B × h = 16 × 9 = 144 cm³.

2. A cylindrical tank has r = 7 m and h = 10 m. Find the volume (π = 22/7).

V = πr²h = (22/7) × 49 × 10 = 22 × 7 × 10 = 1540 m³. That is 1 540 000 litres.

3. A pyramid has a square base of side 6 cm and height 8 cm. Find its volume.

B = 36 cm². V = ⅓ × 36 × 8 = 96 cm³.

4. A cone has base radius 3 cm and slant height 5 cm. Find its volume.

h = √(5² − 3²) = 4 cm. V = ⅓π × 9 × 4 = 12π ≈ 37.7 cm³.

5. Find the volume of a ball of radius 3 cm.

V = (4/3)π × 3³ = (4/3)π × 27 = 36π ≈ 113.1 cm³.

6. A spherical cap of height 2 cm is cut from a ball of radius 5 cm. Find its volume.

V = πk²(3r − k)/3 = π × 4 × (15 − 2)/3 = 52π/3 ≈ 54.5 cm³.

7. A metal ball of radius 6 cm is melted and made into small cones of radius 3 cm and height 4 cm. How many cones are made?

Ball: (4/3)π × 216 = 288π cm³. One cone: ⅓π × 9 × 4 = 12π cm³. Number = 288π ÷ 12π = 24 cones (volume does not change on melting).

Common mistakes

Practice quiz

1. A cone and a cylinder have the same base and height. The cone volume is:
2. The volume of a ball of radius r is:
3. A ball fits exactly inside a cylinder (height 2r). The ball takes what part of the cylinder?
4. Which height is used in V = ⅓πr²h for a cone?
5. A spherical cap with k = r (cap height equals the radius) is a:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

Why is the volume of a cone one third of a cylinder?

The cone slices shrink as you go up. Adding the slices of area πr²(1 − t/h)² over the height gives exactly one third of πr²h. You can also fill a cone with rice three times to fill the matching cylinder.

What is the volume of a spherical segment?

For a cap of height k cut from a ball of radius r, V = πk²(3r − k)/3. A segment between two parallel planes is the difference of two such caps.

Do I need calculus for these volumes?

No. Slicing and Cavalieri's idea are enough for school level, and they are the same idea behind integration. Calculus only makes the adding-up exact for any shape.

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