What Fits Inside

Before you begin

Quiz 27 did the flat shapes: the walk around a garden bed and the soil inside it. This one lifts those shapes off the page. A bed becomes a box, a circle becomes a can, and the two questions from that quiz come back in three dimensions, still two questions and still different from each other.

There is one piece of good news to have in hand before anything else. You are not asked to remember any of these formulas. The GED gives you a formula sheet during the test, and every volume and surface-area formula you could need is printed on it. The sheet is on this site as well, at The Formula Sheet, and it is worth opening in another window while you read this.

So the work in this family is not memory. It is four other things: knowing which of the two questions you are being asked, knowing which solid you are looking at, knowing what each letter in the formula stands for, and keeping the units straight. Everything below is about those four.

Two questions, not one

Stand in front of a cardboard box. There are two entirely different things you might want to know.

What fits inside? That is volume. It is measured in cubes — cubic inches, cubic feet, cubic centimeters — because you are counting how many little cubes would pack the space. Water in a tank, soil in a planter, cereal in a carton: all volume.

What wraps around? That is surface area. It is measured in squares — square inches, square feet — because you are counting the skin. Wrapping paper, paint on a wall, sheet metal to build the box, the label on a can: all surface area.

The whole of volume, in one line

Nearly every volume question on the GED comes from a single idea, and the idea is this: find the area of the base, then multiply by how tall it is. Written short, that is V = Bh, where B is the area of the base and h is the height.

Here is why it works. Take a box 5 units long and 3 units wide. Lay unit cubes across the bottom: you fit 5 × 3 = 15 of them, which is just the area of the base. Now the box is 4 units tall, so you can stack 4 of those layers. Fifteen in a layer, four layers, 60 cubes.

That single idea covers most of the solids on the sheet. What changes from one to the next is only what the base happens to be.

For a box — a rectangular prism, on the sheet — the base is a rectangle, so its area is length times width, and V = lwh. A box 8 inches by 5 by 4 holds 8 × 5 × 4 = 160 cubic inches. You can multiply the three numbers in any order; the answer is the same, which is a useful thing to know when one pair multiplies more easily than another.

For a cylinder — a can, a pipe, a water tank — the base is a circle, and the area of a circle is πr² from Quiz 27. So the volume is V = πr²h. That is the same sentence as before: area of the base, times the height. The formula only looks harder because the base is round.

And for any other right prism — a tent with a triangular end, a pipe with a hexagonal cross-section — the sheet writes it as V = Bh and leaves B to you. Find the area of the end, multiply by the length. The test does not ask this often, but when it does, the formula is telling you to go and work out one flat area first.

A third, for the ones that come to a point

A cone and a pyramid taper. They hold less than the can or the box that would just contain them, and the amount less is exact and surprisingly tidy: a cone holds exactly one third of the cylinder with the same base and the same height. A pyramid holds exactly one third of the matching box.

So both formulas on the sheet are the ones you already have, with a third in front:

cone: V = ⅓πr²h     pyramid: V = ⅓Bh

This is worth believing rather than memorizing, because it turns a hard-looking question into a short one. If a problem tells you a cylinder holds 66 cubic feet and asks for the cone with the same base and height, you do not need π or the radius at all. You divide 66 by 3.

The sphere

A ball has no base and no height, only a radius, and both its formulas use that one number:

V = ⁴⁄₃πr³     SA = 4πr²

The one thing to be careful of is that the volume cubes the radius while the surface area squares it, and it is easy to carry the wrong one across. There is a reason behind it that makes it stick: surface area is a skin, and skins are two-dimensional, so the radius is squared. Volume is a filling, and fillings are three-dimensional, so it is cubed. The exponent is telling you how many dimensions you are measuring.

Surface area: unfold it

Surface area looks like six formulas to learn. It is really one instruction: flatten the solid out and add up the pieces.

Cut a box along its edges and lay it flat and you get six rectangles. They come in three matching pairs — front and back, left and right, top and bottom — and that is exactly what the sheet's SA = 2lw + 2lh + 2wh is saying. Three pairs, each pair counted twice.

A cylinder unfolds too, and this one is worth seeing once because it explains a formula that otherwise looks arbitrary. Cut the top and bottom off a can and you have two circles, each πr². Slit the side and unroll it and you have a rectangle. Its height is the height of the can. Its width is the distance around the circle — the circumference, 2πr. So the side has area 2πrh, and altogether:

SA = 2πr² + 2πrh

Cones and pyramids have a wrinkle worth naming. Their sheet formulas use s, the slant height — the distance up the sloping outside, not the straight-up height h in the middle. A cone is SA = πr² + πrs: the circle on the bottom, plus the curved side. If a question gives you h and wants surface area, it is quietly asking you to find s first, and the tool for that is the Pythagorean theorem, because r, h and s make a right triangle. That is a hard question and the GED does not ask it often, but when a cone problem looks impossible, this is usually why.

The sheet is in the room

Since you are not being tested on memory, it is worth being deliberate about what you are being tested on. Work in this order.

One. Read the question and decide: inside, or around? Water, air, soil, sand, how much it holds, capacity — volume. Paint, paper, metal, label, cover, how much it takes to make — surface area.

Two. Name the solid. Box, can, ball, cone, pyramid, or a prism with some other end.

Three. Find that line on the sheet and write it down before you put any numbers in. Copying the formula onto your scratch paper takes four seconds and it stops the most common kind of error, which is substituting into a formula you are half remembering.

Four. Check what each letter needs. This is where r against d lives, and it is where most of the lost points in this family are.

Five. Substitute, calculate, and write the unit.

The guard

Four things go wrong in this family, and the same four every time.

Radius against diameter. A can is described as 10 inches across. That is the diameter. Every formula wants the radius, which is 5. Putting 10 where r belongs does not make the answer twice too big — because r is squared, it makes it four times too big. Cross out the diameter on your paper and write the radius beside it before you start.

Cubic against square. Said above and said again here, because it is the check that catches everything else.

Mixed units. A tank given as 3 feet wide and 18 inches deep has to be made one thing or the other before you multiply. That is Quiz 26's work and it comes back here constantly.

What to do with π. Unless the question says otherwise, use 3.14 — the sheet gives that value. But read the answer choices first. If they are written as 45π and 90π, the test wants the exact form and you should not multiply the π out at all; just carry it along like a unit and match.

Three to study before you start

Now you

Work on paper. A calculator is fine for the arithmetic; the thinking is in choosing the formula. Use π ≈ 3.14 where you need it, and write the unit on every answer. Questions 6 and 10 have two parts.

1. A storage box is 8 inches long, 5 inches wide and 4 inches tall. What is its volume?
2. What is the surface area of that same box?
3. A cylinder has a radius of 5 cm and a height of 12 cm. What is its volume?
4. A cylindrical can is 10 inches across the top and 6 inches tall. What is its volume?
   • A) 94.2 cubic inches
   • B) 188.4 cubic inches
   • C) 471 cubic inches
   • D) 1884 cubic inches
5. A cylinder holds 66 cubic feet. A cone has the same base and the same height as that cylinder. How much does the cone hold?
6. A shipping box measures 2 feet by 2 feet by 3 feet. (a) What is its volume? (b) How much wrapping paper would cover it exactly, with no overlap?

7. A ball has a radius of 3 inches. What is its volume?
8. Dev says that a can twice as tall holds twice as much, and a can twice as wide also holds twice as much. He is right about one of these and wrong about the other. Say which is which, and why.
9. A rectangular swimming pool is 25 meters long, 10 meters wide and 2 meters deep. Which of these is the amount of water needed to fill it?
   • A) 250 cubic meters
   • B) 500 square meters
   • C) 500 cubic meters
   • D) 640 square meters
10. A soup can has a radius of 4 cm and a height of 11 cm. (a) How much soup does it hold? (b) The paper label wraps around the side only, top to bottom, with no overlap. What is the area of the label?

Check your work

The ball in the can

In 212 BC a Roman army under Marcellus took the Greek city of Syracuse, in Sicily, after a siege of two years. The siege had lasted that long in part because of one man. Archimedes had spent his old age building the machines that defended the harbor: cranes that reached over the walls and lifted ships out of the water, catapults ranged for every distance an attacker might stand. The Romans had come to dread them. When the city finally fell, Marcellus gave an order that the old man was not to be harmed.

A soldier found him working. The accounts differ on what happened next — Archimedes was drawing a figure in the dust, or in the sand of a tray, and asked to be allowed to finish it, or refused to leave until the problem was solved. The soldier ran him through. Marcellus, the sources agree, was furious, and had him buried with honor.

Here is the part that matters. Archimedes had left an instruction about the grave. He did not want a ship-lifting crane on it, or a catapult, or any record of the two years he had held off an empire. He wanted a ball inside a can: a sphere drawn inside the cylinder that just contains it, with a number written beside the figure.

The number was two to three.

He had proved that a sphere takes up exactly two thirds of the space of the cylinder that fits snugly around it — and, more remarkably, that its curved skin is also exactly two thirds of that cylinder's whole surface. The same simple fraction, twice, for two things that have no obvious reason to agree. You can check the first with what is on the formula sheet. A sphere of radius r has volume ⁴⁄₃πr³. The cylinder around it has that same radius and a height of 2r, so its volume is πr² × 2r = 2πr³. Divide one by the other and the π and the r³ both cancel, leaving ⁴⁄₃ ÷ 2. Dividing by 2 is halving, and half of four thirds is two thirds. Whatever the size of the ball, the ratio never moves.

He had no algebra to do this with. The symbols in this quiz would not exist for another eighteen centuries. He had no calculus either, which is the tool a student would reach for today. What he had was a method of his own: to find a curved thing, box it in with straight things from the outside and fill it with straight things from the inside, then make both sets finer and finer until the gap between them closed. It is called the method of exhaustion, and it is slow, and it is exactly right. He considered it his best work.

Then the tomb was forgotten. Syracuse kept it for a while and stopped keeping it.

In 75 BC, a hundred and thirty-seven years later, a young Roman official named Cicero was posted to Sicily as quaestor — a financial administrator, the lowest rung of a career he intended to climb. He asked the people of Syracuse where Archimedes was buried. They told him there was no such tomb. He did not believe them. He went out to the gate where the old graves were, through ground thick with brambles, and worked along the overgrown stones until he saw a shape cut into one of them above the scrub: a sphere, and a cylinder around it.

He had the brush cleared. The inscription was worn but the figure was not, and the verses could still partly be read. Cicero wrote about it years later with a plain kind of anger: that the most famous city in Greek Sicily, once a great seat of learning, had not known where its greatest man lay, and that it took a visitor from a town of no distinction to find him under a bush.

Sources: Plutarch's Life of Marcellus for the siege and the death; Cicero, Tusculan Disputations V, for the finding of the tomb; Archimedes' own On the Sphere and the Cylinder for the proof.