Solids: Volume, Sections & Rotations

Every solid here is a flat shape that has been stacked, tapered or spun.

Take a rectangle and spin it about one of its edges. What does it sweep out?

The edge stays put. The far side traces a circle. Every point in between traces a circle too.

spin a flat shape and it sweeps out a solid a rectangle a cylinder spin a right triangle instead and you get a cone

The result is a cylinder — and its radius is the rectangle's width, its height the rectangle's height.

Spin a right triangle about a leg instead and you get a cone. Spin a semicircle about its diameter and you get a sphere.

Three solids, three flat shapes, one idea.

And it runs the other way too. Slice a solid and you get a flat shape back — which is why cross sections and solids of revolution are two halves of the same unit.

spin a flat shape and it sweeps out a solid a rectangle a cylinder spin a right triangle instead and you get a cone

Stacked solids: prisms and cylinders

A prism is a base shape stacked straight up. Its volume is the area of the base times the height: V = Bh.

A cylinder is the same idea with a circular base, so V = π r2 h — it is not a separate formula, just Bh with B = π r2.

Because of Cavalieri, this works even if the stack leans over. Only the perpendicular height counts.

One formula, many solids

V = Bh covers every prism and cylinder. Find the area of the base and multiply by the height — the shape of the base is the only thing that changes.

Tapered solids: pyramids and cones

Taper a prism to a point and you get a pyramid, which holds exactly one third as much: V = 13 Bh.

A cone is a pyramid with a circular base, so V = 13 π r2 h.

The 13 is the same one you can pour — three coneloads fill the matching cylinder.

Spheres

V = 43 π r3 and surface area = 4π r2.

The surface area is worth noticing: it is exactly four times the area of the circle you would get by slicing through the middle.

A sphere holds two thirds of the cylinder that just contains it — Archimedes' result, and the one on his tomb.

A sphere sliced through the middle gives a circle of area πr²; the surface area is four of those circles, and the sphere holds two thirds of the cylinder around it slice through the middle: that circle, four times over, is the surface r the middle slicearea πr² πr²πr²πr²πr² surface area = 4πr² volume ⁴⁄₃πr³= ⅔ of the cylinderthat just contains it cylinder: πr² × 2r = 2πr³, and ⅔ of 2πr³ is ⁴⁄₃πr³

Cross sections

Slicing a solid exposes a flat shape, and which shape depends on the angle of the cut.

A cut parallel to a cylinder's base gives a circle; a vertical cut through the axis gives a rectangle; a slanted cut gives an ellipse.

A cube can be sliced to give a triangle, a rectangle, a pentagon or a regular hexagon — the solid never changed, only the plane.

Surface area

Surface area is the total of every face. The reliable method is to unfold the solid into its net and add up the flat pieces.

A cylinder unrolls into two circles and one rectangle whose width is the circumference: SA = 2π r2 + 2π r h.

Volume is in cubic units and surface area in square units, so the units on your answer say which one you found.

Slant height, not vertical height

The lateral surface of a cone uses the slant height ℓ, not the vertical height h. They are different, and related by ℓ2 = r2 + h2.

Worked examples

Prism volume

A prism has a triangular base with legs 6 and 8, and a height of 15. Find its volume.

  1. Base area first: B = 12 (6)(8) = 24 A right triangle.
  2. V = Bh = 24 × 15 Stacked solid, so base times height.
  3. V = 360 cubic units Cubic units, because it fills space.

360 cubic units

Cone surface area

A cone has radius 5 and vertical height 12. Find its total surface area in terms of π.

  1. Slant height: ℓ = √52 + 122 = 13 Lateral area needs ℓ, not h.
  2. Lateral area = π r ℓ = π(5)(13) = 65π The curved surface.
  3. Base = π r2 = 25π The circle underneath.
  4. Total = 65π + 25π = 90π A 5-12-13 triangle made the slant height clean ✓

90π square units

A solid of revolution

A right triangle with legs 3 and 4 is spun about the leg of length 4. Describe the solid and find its volume in terms of π.

  1. Spinning a right triangle about a leg gives a cone. The axis leg becomes the height.
  2. The axis is the leg of 4, so h = 4; the other leg sweeps the base, so r = 3. Which leg you spin about decides which is which.
  3. V = 13 π (3)2 (4) = 13 π (36) Substitute.
  4. V = 12π Spinning about the other leg would give 16π — a different cone.

A cone; 12π

Words to know

Prism
Two congruent parallel bases joined by rectangles. V = Bh.
Pyramid
A base tapering to a point. V = 13 Bh.
Cross section
The flat face exposed by slicing a solid.
Solid of revolution
The solid swept out by spinning a flat shape about a line.
Lateral area
The surface area not counting the bases.
Cavalieri's principle
Same height and same cross sections at every level means same volume.

Mistakes to watch for

Using the vertical height where the slant height belongs. A cone's lateral area needs ℓ, and ℓ = √r2 + h2 is always longer than h. Using h makes the surface come out too small.

Forgetting the 13 on pyramids and cones. Tapered solids hold a third of the prism or cylinder that wraps them — you can pour it three times to check, and without the third the answer is triple.

Mixing volume and surface area. Volume fills and is measured in cubic units; surface area wraps and is measured in square units. The units on your answer identify which one you actually computed.

Assuming every cross section matches the base. A slanted cut through a cylinder gives an ellipse, not a circle, and a cube can be cut into a hexagon. The angle of the plane decides the shape, not the solid alone.

Cavalieri's principle, from 1635, says two solids with the same height and the same cross-sectional area at every level have the same volume.

So a leaning stack of coins holds exactly as much as a straight one — which is why an oblique prism uses the same formula as a right one.

It was one of the ideas that led directly to integral calculus fifty years later.

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