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3D objects generated by rotations of 2D figures

20 practice questions 2 video lessons Theory + worked examples

Solids of Revolution

California Geometry • Standard G-GMD.4 • Three-Dimensional Measurement

Solids of Revolution is a topic in Three-Dimensional Measurement in the California Common Core State Standards. It is aligned to Standard G-GMD.4, which requires students to identify three-dimensional objects generated by rotations of two-dimensional objects.

A solid of revolution is generated by rotating a two-dimensional region about an axis — a rectangle sweeps a cylinder, a right triangle a cone.

California Geometry › Three-Dimensional Measurement › Solids of Revolution  —  Standard G-GMD.4

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Practice questions

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  • Rotating 2D shapes in 3D | Perimeter, area, and volume | Geometry | Khan Academy Watch
  • Rotations of 2D Shapes Watch
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Theory

Rotating a 2D shape a full turn about an axis sweeps out a solid of revolution:

  • Rectangle about a side \(\to\) cylinder.
  • Right triangle about a leg \(\to\) cone.
  • Semicircle about its diameter \(\to\) sphere.
The axis becomes the center line; the far edge of the shape sweeps the outer surface.
Rotating a rectangle gives a cylinder Rotating a rectangle about one side sweeps out a cylinder. axis → rectangle cylinder
Rotating a rectangle about a side sweeps out a cylinder.
Solids of revolution Solids of revolution Solids of revolution rectangle → cylinder right triangle → cone semicircle → sphere
Common solids of revolution.

The rotations:

\[\text{rectangle}\to\text{cylinder},\quad \text{right triangle}\to\text{cone},\quad \text{semicircle}\to\text{sphere}\]
a rectangle rotates to a cylinder, a right triangle to a cone, a semicircle to a sphere
The distance from the axis to the far edge becomes the radius of the solid.

How to find the solid of revolution

  1. Identify the 2D shape and the axis of rotation.
  2. Sweep the shape a full turn about the axis.
  3. Name the resulting solid.
  4. Read the radius as the distance from the axis to the farthest point.
Example 1 — Rotate a rectangle
What solid forms when a rectangle is rotated about one of its sides?
Solution

A cylinder — the side is the axis, the opposite side sweeps the curved surface.

a rectangle rotated about a side forms a cylinder
Example 2 — Rotate a right triangle
What solid forms when a right triangle is rotated about a leg?
Solution

A cone — the leg is the axis and the hypotenuse sweeps the slanted surface.

a right triangle rotated about a leg forms a cone
Example 3 — Rotate a semicircle
What solid forms when a semicircle is rotated about its diameter?
Solution

A sphere.

a semicircle rotated about its diameter forms a sphere
Example 4 — Volume of the swept solid
A rectangle \(3\) wide and \(7\) tall is rotated about its height. Find the volume of the cylinder (leave \(\pi\)).
Solution

The width becomes the radius (\(3\)) and the height stays \(7\).

\(V\)\(=\)\(\pi(3)^2(7)=63\pi\)
the volume is 63 pi

Common pitfalls

The axis matters. Rotating about a different side or leg changes the solid's dimensions.
The distance to the axis is the radius, not the whole width.
A semicircle makes a sphere; a full circle would overlap itself.

Frequently asked questions

What is a solid of revolution?

A 3D solid formed by rotating a 2D shape a full turn about an axis.

What solid does a rotating rectangle make?

A cylinder, when rotated about one of its sides.

What solid does a rotating right triangle make?

A cone, when rotated about one of its legs.

What solid does a rotating semicircle make?

A sphere, when rotated about its diameter.

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