A right solid is a three dimensional geometric object with a base that is either a circle or a regular polygon. It may come to a point or have a flat top. The flat top must be identical and parallel to the base, and the sides are then perpendicular to them. If instead the solid is pointed, a line from the point to the center of the base must be perpendicular to the base. These objects make up the geometric categories of the pyramid, prism, cylinder and cone. Their volumes are proportional to the area of the base multiplied by the height.

If the base of the object is round, calculate the area of this circle by squaring the radius (or squaring the diameter and dividing by four). Multiply the result by Pi (approximately 3.14). This is the area of the circular base of the cylinder or cone.

If the base of the object is an equilateral triangle, calculate its area by multiplying the length of one side of the triangular base by the square root of 3 and then divide by 4. This is the area of the base of the three-sided pyramid or prism.

If the base is a square, find its area by multiplying the length of the side by itself (squaring it). This is the area of the base of the square pyramid or prism.

Multiply the area of the base by the height of the solid.

If the solid is a prism or a cylinder this result is the volume. Prisms and cylinders have tops and bottoms that are parallel to each other and sides that are perpendicular to the two ends. Prisms have polygon bases while cylinders are round.

For example, a prism has a square base that is 8 inches by 8 inches and it is 6 inches high. The area of the base is 8 inches squared or 64 square inches. The volume is 6 inches times 64 square inches or 384 cubic inches.

If the solid is a pyramid or a cone, divide the result of step 4 by three to find the volume. Pyramids have polygons for bases, and cones are round. Both types of objects have side surfaces that come to a point rather than having flat tops.

For example, a cone is 4 inches high and has a base that is 10 inches across. Its radius is 10 divided by 2 equals 5 inches, so its area is 5 squared times Pi which is approximately 3.14 times 25 or 78.54 square inches. The volume is 4 inches times 78.54 square inches divided by 3 which is about 104.72 cubic inches.

#### References

- “CRC Standard Mathematical Tables and Formulae”; Daniel Zwillinger; 2002

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