Definition:Solid of Revolution

Definition

Let $F$ be a plane figure.

Let $L$ be a straight line in the same plane as $F$.

Let $F$ be rotated one full turn about $L$ in $3$-dimensional space.

The solid of revolution generated by $F$ around $L$ is the solid figure whose boundaries are the paths taken by the sides of $F$ as they go round $L$.

The usual scenario is that:

$(1): \quad$ one of the sides of $F$ is a straight line coincident with $L$

$(2): \quad$ $L$ itself is aligned with one of the coordinate axes, usually the $x$-axis in a cartesian plane

$(3): \quad$ Two other sides of $F$ are also straight lines, perpendicular to $L$

$(4): \quad$ The remaining side (or sides) of $F$ are curved and described by means of a real function.

The above diagram shows a typical solid of revolution.

The plane figure $ABCD$ has been rotated $2 \pi$ radians around the $x$-axis.

$FECD$ illustrates the disposition of $ABCD$ part way round.

Axis of Revolution

The straight line around which the rotation is being performed is known as the axis of revolution.

Also known as

A solid of revolution is also known as:

a solid of rotation

a volume of rotation.

Also see

• Definition:Surface of Revolution

• Results about solids of revolution can be found here.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Area, Volume and Centre of Gravity: Volume of Rotation about the $x$-axis
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): revolution
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): solid of revolution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): revolution
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): solid of revolution