Combined Volume and Surface Area

What fills it and what covers it

A solid can be measured in two different ways. Its volume is the space inside it, which is how much it holds. Volume is counted in cubes 1 cm along each edge, so it is measured in cubic centimeters, cm³. Its surface area is the total area of all its faces, which is how much material it takes to cover it. Surface area is counted in squares 1 cm along each side, so it is measured in square centimeters, cm².

A can of soup has both. The soup inside the can fills its volume. The metal of the can, its top, its bottom and its curved side together, makes up its surface area.

The volume: the circle times the height

Take a can with a radius of 3 cm and a height of 10 cm. A cylinder is shaped like a prism: every cut straight across it uncovers the same circle. So its volume is the area of that circle times the height, $V = \pi r^{2}h$.

The circle has a radius of 3 cm, so its area is $\pi \times 3^{2} = 9\pi$ cm². Each centimeter of height holds a layer of $9\pi$ cm³, and the cylinder is 10 cm tall, so its volume is $9\pi \times 10 = 90\pi$ cm³.

The can is drawn lying on its side. Every cut across it uncovers the same circle, of radius 3 cm and area $9\pi$ cm², and the can is 10 cm from end to end: $9\pi \times 10 = 90\pi$ cm³.

The surface area: lay it flat

To find what covers the cylinder, imagine cutting it open and laying every face flat. This flat shape is the net of the cylinder. It has two circles, the top and the bottom, and one rectangle, which is the curved side unrolled.

The rectangle is as tall as the cylinder, 10 cm. Its width is the distance around the circle, the circumference: $2\pi r = 2 \times \pi \times 3 = 6\pi$ cm. So the curved side has an area of $6\pi \times 10 = 60\pi$ cm².

Each circle has an area of $\pi \times 3^{2} = 9\pi$ cm². Add all three faces: $9\pi + 9\pi + 60\pi = 78\pi$ cm². For any closed cylinder, the surface area is $2\pi r^{2} + 2\pi r \times h$.

The net of a cylinder: two circles of area $\pi r^{2}$, and a rectangle whose width is the circumference $2\pi r$ and whose height is the height of the cylinder, h.

In decimals, and in milliliters

The answers $90\pi$ cm³ and $78\pi$ cm² are exact, because they keep $\pi$ as a symbol. For a decimal, multiply by $\pi$, which is about 3.14159. The volume is $90\pi \approx 282.7$, which is about 283 cm³ to the nearest whole number. The surface area is $78\pi \approx 245.0$, which is about 245 cm².

A milliliter is the same amount of space as a cubic centimeter: 1 ml = 1 cm³. So a can with a volume of about 283 cm³ holds about 283 ml of soup.

Two measures that cannot be compared

The can has a volume of $90\pi$ cm³ and a surface area of $78\pi$ cm². The two numbers count different things, cubes and squares, so it makes no sense to ask which one is bigger. Measure the same can in millimeters instead. A cubic centimeter is 10 × 10 × 10 = 1000 mm³, and a square centimeter is 10 × 10 = 100 mm², so the volume becomes $90{,}000\pi$ mm³ and the surface area becomes $7800\pi$ mm². In centimeters the volume’s number was a little larger, 90 against 78; in millimeters it is more than ten times larger. Keep each answer with its own unit: cm³ for a volume and cm² for an area.

Three slips

Giving $78\pi$ as the volume answers the wrong question: $78\pi$ cm² is what covers the can, not what fills it.

The curved side alone, $60\pi$ cm², is an area. Its unit is cm², so it cannot be the volume, and it is not the whole surface either.

A closed can has a top and a bottom. Counting one circle gives $9\pi + 60\pi = 69\pi$ cm², which is the surface of a can with no lid. Read the question to see which faces are there.

A solid made of two pieces

Many real objects are two solids joined together, such as a cylinder standing on a cone. Split the object where the two pieces meet, and work with each piece on its own.

Volumes add, because the two pieces fill separate spaces. Surface areas do not simply add. Where the pieces meet, a face of each one is pressed against the other, so those two faces are inside the object and are not part of its surface. Add only the outside surfaces.

For a cone, three facts are needed. Its volume is $\frac{1}{3}\pi r^{2}h$, a third of the cylinder with the same base and height. Its curved surface has area $\pi rl$, where l is the slant height, the length of the sloping side from the tip to the rim. The radius, the height and the slant height make a right-angled triangle inside the cone, so $l^{2} = r^{2} + h^{2}$ by Pythagoras.