Solids of Revolution
Take a flat shape, pick a line, and spin the shape all the way around that line. The path it sweeps out is a three-dimensional solid — a solid of revolution. It is how a potter's wheel turns a flat profile into a vase.
The trick is to picture the spin. Every point on the flat shape traces a circle around the axis, so the solid is always round when viewed down the axis. Which round solid you get depends on the flat shape.
The three basic spins
Spin a rectangle about one of its sides and it sweeps out a cylinder — the side is the axis, and the opposite side traces the circular wall. Spin a right triangle about one leg and you get a cone, with the leg as the axis and the hypotenuse tracing the slanted surface.
Spin a semicircle about its diameter and every point traces a circle, filling in a sphere. These three cover most of what you will be asked to identify.
The figure shows the cone case: the dashed line is the axis, and the right triangle's hypotenuse sweeps out the slanted surface.
Why the axis matters
The same shape gives different solids depending on which line you spin it around. A rectangle spun about a side gives a solid cylinder; spun about a line beside it (not touching), it gives a cylinder with a hole — a washer shape.
Always locate the axis first. Every point's distance from that axis becomes a radius in the finished solid.
Worked examples
Example 1: a spinning rectangle
A rectangle is rotated about one of its sides. What solid forms?
Answer: A cylinder
Example 2: a spinning right triangle
A right triangle is spun about one of its legs. What solid forms?
Answer: A cone
Example 3: a spinning semicircle
A semicircle is rotated about its straight diameter edge. What solid forms?
Answer: A sphere
Try one yourself
Common questions
What shape spins into a sphere?
A semicircle rotated about its straight diameter edge. Every point on the arc traces a circle, and together they fill a complete sphere.
Does the shape have to touch the axis?
No. If the shape is set away from the axis, the spin leaves a hole in the middle — like spinning a small rectangle beside a line to make a ring-shaped solid.
How is this used in real design?
Anything made on a lathe or potter's wheel is a solid of revolution — bottles, table legs, bowls. The designer draws the flat profile and the machine spins it into the 3D part.
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