Volumes of Pyramids and Cones: Practice

The volume determines the perpendicular height.

This gives fifteen centimeters. Its square plus the offset square is $225+64$, which equals the square of seventeen. Both records therefore support $15$ cm.

Use the volume to find the radius square.

This gives sixty-four. For the slant height, add the radius and height squares.

The section has one third of the base lengths and one ninth of its area. Thus the base area is $180$ square units.

The volume is $720$ cubic units.

The cone volume is fifty-six times pi cubic centimeters.

Set the pyramid volume equal to it and solve.

The center-to-midpoint distance is five centimeters. In right triangle SOM, the thirteen-centimeter face altitude is the hypotenuse.

The square base area is one hundred square centimeters.

The volume is $400$ cubic centimeters.

View BCF and BEF as bases in the same plane, with common apex A. Their perpendicular heights agree. Each parallel section uses the same squared height fraction of equal base areas, so Cavalieri gives equal volumes. Neither base perimeter nor congruence is required, and one sampled section alone is insufficient.

The section-to-base radius ratio equals the apex-distance fraction.

Solving gives a whole height of $12$ cm, measured from the apex to the base plane.

First recover the full base area.

The base area is thirty-two square centimeters. Dividing by the known four-centimeter side gives the other side, $8$ cm.

Both sections are halfway from apex to base plane. Their radii are therefore half the base radius, even though their centers can be shifted.

Both section areas are $9\pi/4$ square centimeters.

Use the volume to recover the whole perpendicular height.

The apex-distance fraction is one third, so the area is one ninth of the base disk area.

Recover the triangular base area from the volume.

This is twenty-four square centimeters. Let b be the requested edge length.

The volume gives the whole height.

The height is ten centimeters. The section radius is half the base radius, so the section is five centimeters below the apex. Subtract from the whole height to get $5$ cm above the base plane.