Key takeaway: A solid of revolution is a three-dimensional shape created by rotating a flat region about a line, called the axis, so that every point of the region sweeps out a circle.
Grade 10 – GeometryG-GMD.B.4Geometry and Measurement
Definition of Solid of Revolution
A solid of revolution is the shape you see when a flat figure spins around a straight line. A rectangle spun about one of its sides leaves a cylinder. A right triangle spun about one of its legs leaves a cone. A half-circle spun about its diameter leaves a sphere.
Each point of the flat region moves along a circle centered on the axis, so slices taken perpendicular to the axis are circles or rings. The choice of axis matters a great deal: the same triangle spun about its two different legs gives two different cones.
Formula and notation
For a solid of revolution built from a right triangle with legs \(r\) and \(h\) spun about the leg of length \(h\), the result is a cone with radius \(r\) and height \(h\):
\[ V = \frac{1}{3}\,\pi r^2 h \]
A rectangle \(r\) wide and \(h\) tall spun about its \(h\)-side gives a cylinder, \(V = \pi r^2 h\).
Key facts
- Every slice perpendicular to the axis is a circle or a ring.
- A rectangle gives a cylinder, a right triangle a cone, a semicircle a sphere.
- The leg on the axis becomes the height; the leg off the axis becomes the radius.
- Different axes for the same shape produce different solids.
Solid of Revolution example
A right triangle has legs of 6 cm and 15 cm. Find the volume of the solid made by spinning it about the 15 cm leg, and compare it with the solid made by spinning it about the 6 cm leg.
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About the 15 cm leg: radius 6, height 15, so \(V = \frac{1}{3}\pi(6)^2(15) = 180\pi \approx 565.5\) cm³.
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About the 6 cm leg: radius 15, height 6, so \(V = \frac{1}{3}\pi(15)^2(6) = 450\pi \approx 1{,}413.7\) cm³.
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Compare: \(\frac{450\pi}{180\pi} = 2.5\).
The volumes are \(180\pi \approx 565.5\) cm³ and \(450\pi \approx 1{,}413.7\) cm³; the second cone holds 2.5 times as much.
Common mistake
Watch out
Joel says spinning the triangle about either leg gives the same cone, since it is the same triangle.
Why it is wrong
The axis decides which leg becomes the height and which becomes the radius. A cone 15 cm tall with a 6 cm radius is long and thin, and one 6 cm tall with a 15 cm radius is wide and flat. The volumes differ, as 180π and 450π show.
Related terms and lessons
- Volume of a Cone
- Volume of a Cylinder
- Volume of a Sphere
- Cross Section
- Hypotenuse
- Cavalieri's Principle
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