Solids, Volume and Surface Area

There look to be seven formulas here and there are really two, plus a factor of one third and one genuine oddity.

The one that covers most of it

Enlarge

Zoom in

Volume = area of the base x height.

This holds for every solid with the same cross-section all the way up: a box, a cylinder, a triangular prism, a length of pipe.

Learning it this way rather than as separate formulas means a prism with an odd base is not a new problem. Work out the area of the end face, multiply by the length.

The height is measured perpendicular to the base, the same rule as in flat geometry. For a cylinder lying on its side, the "height" is its length.

The one third

Cone: one third of the cylinder with the same base and height.

• Cylinder: pi x r squared x h.
• Cone: one third x pi x r squared x h.

Pyramid: one third of the prism with the same base and height.

• Prism: base area x h.
• Pyramid: one third x base area x h.

That is the only thing to remember. A cone with a radius of 3 and a height of 10 holds one third of pi times 9 times 10, which is 30 pi.

Forgetting the one third gives exactly three times too much, and that value is always among the choices.

The sphere

The one that does not follow a pattern, so it has to be memorized.

• Volume: four thirds x pi x radius cubed.
• Surface area: 4 x pi x radius squared.

A sphere of radius 3 has volume four thirds times pi times 27, which is 36 pi.

The only real defense against mixing these two up is the exponent. Volume is cubed, surface area is squared, and that is true of every solid, not just spheres. If your volume formula has a square in it, something is wrong.

Surface area

Surface area is the total of every face, so for a box it is the sum of six rectangles. Opposite faces match, so:

Box: 2lw + 2lh + 2wh.

Cube: 6 times side squared, since all six faces are identical.

Cylinder: two circular ends plus the tube, which unrolls into a rectangle whose width is the circumference:

2 x pi x r squared, plus 2 x pi x r x h.

The unrolling is worth picturing once. The side of a can, cut and laid flat, is a rectangle as tall as the can and as wide as the way around it. Nothing needs memorizing after that.

Capacity and units

Volume questions often finish in gallons or liters rather than cubic units, and that conversion is where points go.

Two worth knowing:

• 1 cubic foot is about 7.48 gallons.
• 1 liter is 1,000 cubic centimeters, and 1 milliliter is 1 cubic centimeter.

And the rule from unit conversion applies with force here: a cubed unit takes the conversion factor cubed. One cubic yard is 27 cubic feet, not 3, because three feet convert in each of three dimensions.

Scaling

Doubling a tank's dimensions makes it hold eight times as much. This is asked directly and the factor-of-2 answer is always a choice.

What you can skip

Across the 43 questions on this topic:

• Frustums and Euler's formula never appear.
• Lateral area as a separate formula comes up three times. Lateral area is the surface area without the top and bottom.

Where people lose points

Dropping the one third on a cone or a pyramid.

Using the diameter instead of the radius in a cylinder or a cone, which makes the answer four times too large because the radius is squared.

Mixing up the sphere formulas. Check the exponent.

Reporting a volume in square units or a surface area in cubic ones.

Converting a cubic unit with the plain factor.

Using a slant height as the height on a cone or a pyramid. The height is the perpendicular distance from the base to the apex; the slant height runs up the outside and is longer.

Work one in under a minute

A cylindrical tank is 6 feet across and 10 feet tall. How many cubic feet does it hold, to the nearest whole number?

Diameter 6 means radius 3.

Base area: pi times 9, about 28.27.

Volume: 28.27 times 10 is about 283 cubic feet.

Check: a 6 by 6 by 10 box holds 360, and a cylinder fills a bit under 79 percent of its bounding box, so 283 is right. Had the answer come out near 1,130, that would be the diameter used as the radius.

Where this leads

Volume questions are the most common measurement shape in Arithmetic Reasoning, and the base-area idea is what makes an unfamiliar prism answerable.