Problem

A filled right triangle has perpendicular legs \(a\) and \(b\) and hypotenuse \(c\). It rotates \(360°\) about leg \(a\). Use the diagram to identify the solid and give its radius, height, slant height, and volume.

Prompt diagram for M3-037-A07-V01: the generating region, rotation axis, dimensions, and 360-degree sweep with the resulting solid withheld.

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What this problem is really about

Map each side of the right triangle by its relation to the rotation axis. The leg on the axis becomes the solid's axial dimension, the perpendicular leg sweeps the base radius, and the hypotenuse sweeps the sloping boundary. Because the whole triangular region rotates, the result is filled; use the mapped radius and height in the appropriate volume formula.

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The result is a hollow conical surface with radius b, height a, slant height c, and enclosed volume 0.

The result is one right circular cylinder with radius b, height a, and volume pi b² a.

The result is one right circular cone with radius b, height a, slant height c, and volume (1/3)pi b² a.

The result is one right circular cone with radius a, height b, slant height c, and volume (1/3)pi a² b.

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A filled right triangle has perpendicular legs \(a\) and \(b\) and hypotenuse \(c\). It rotates \(360°\) about leg \(a\). Use the diagram to identify the solid and give its radius, height, slant height, and volume.

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