Solid of Revolution
In short: A solid of revolution is the three-dimensional shape swept out when a flat region spins a full turn around a fixed line, called the axis of rotation.
GeometryGeometryG-GMD.B.4
What is Solid of Revolution?
A solid of revolution is the ball-like blur you see when a coin spins on a table: take a flat shape, choose a line as an axis, and rotate the shape \(360^{\circ}\) around it. Every point travels along a circle, and the solid is the union of those circles.
Three classics: a right triangle turned about one leg makes a cone, a rectangle turned about one side makes a cylinder, and a semicircle turned about its diameter makes a sphere. Slices taken perpendicular to the axis are always circles or rings.
How to spot it
- A problem says a region is rotated, spun or revolved about a line, producing a solid of revolution.
- A lathe, pottery wheel or turned wooden leg produces a rounded object from a flat profile.
- A figure shows a flat shape next to a dashed axis, with an arrow marking a full turn.
- The volume of a cone, cylinder or sphere is wanted, and the original shape is a triangle, rectangle or semicircle.
The method
- To build a solid of revolution, sketch the flat region and the axis, and decide which side of the shape touches the axis.
- Identify the solid: triangle gives a cone, rectangle gives a cylinder, semicircle gives a sphere.
- Read the radius as the distance from the axis, and the height as the length along the axis.
- Apply the matching volume formula and state the answer in cubic units.
Solid of Revolution: solved example
Problem
A right triangle has legs of 4 cm and 6 cm. It is spun a full turn around its 6 cm leg. Describe the solid and find its volume.
Solution
The 6 cm leg lies on the axis, so it becomes the height, and the 4 cm leg sweeps out a circular base of radius 4. The solid is a cone.
\(V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(4)^2(6) = 32\pi \approx 100.5\). The cone’s volume is \(32\pi \approx 100.5\) cubic cm.
Try it
Question 1
A semicircle of radius 5 is spun around its straight edge. Name the solid and find its volume.
Show the answer
It makes a sphere: \(V = \frac{4}{3}\pi(5)^3 = \frac{500\pi}{3} \approx 523.6\) cubic units.
Question 2
A rectangle 2 inches wide and 9 inches long is spun around one of its 9-inch sides. Find the volume of the solid.
Show the answer
The result is a cylinder with radius 2 and height 9: \(\pi(2)^2(9) = 36\pi \approx 113.1\) cubic inches.
Solid of Revolution vs. Cross Section
A cross-section starts with a solid and cuts it open to reveal a flat shape. This construction runs the other way: it starts with a flat shape and spins it to build a solid. A cone is a good bridge, since slicing it perpendicular to its axis gives a circle, and the circle’s radius comes from the original triangle.
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