Circles & Equations

You already know how equations work. You have known it since you learned the times tables. So we start small, with something you know.

Two rows of beads, three beads in each row. How many beads? 2 × 3 = 6

Now read the same picture backward. You know there are 6 beads, and they sit in rows of 3. How many rows? 6 ÷ 3 = 2

And if you know there are 6 beads in 2 rows, how many are in each row? 6 ÷ 2 = 3

One picture. Three facts. Whenever two numbers multiply to make a third, all three facts come together, as a family:

Factors and the product

These three numbers have names. The numbers you multiply are called factors. The answer you get is called the product.

In 2 × 3 = 6, the factors are 2 and 3. The product is 6.

A number is a factor or a product only inside one multiplication

“Factor” and “product” are not labels a number carries everywhere it goes. They name what a number is doing in one particular multiplication: how it stands with the other two numbers in that fact.

In the first fact, 2 and 3 are multiplied to get 6, so 6 is the product. In the second fact, 6 is multiplied by 5 to get 30. Now 6 and 5 are the factors, and 30 is the product. Same number, different fact, different name. Watch the color of the 6 change.

Change the rows and the beads in each row. Then tap each of the three facts to see it in the picture.

The numbers can be anything. The names can be anything. The relationship among the three stays the same.

Two factors multiply to make a product:

factor × factor = product

Then, just as with the beads, each factor is the product divided by the other factor:

one factor = productthe other factor

Here it is with words nobody has ever heard before:

Kuku × Shashima = Petromakrilash

Kuku and Shashima are the factors. Petromakrilash is the product. You don’t know what a Kuku is. You don’t need to. The relationship tells you everything:

Kuku = PetromakrilashShashima

Shashima = PetromakrilashKuku

A picture that holds all three

Draw a triangle and cut it in three. The product goes in the top. The two factors sit side by side at the bottom.

Now cover the one you want to find. What is left tells you what to do:

• Side by side means multiply.
• One above the other means divide, the top one over the bottom one.

Tap any part of the triangle to cover it. Work through the examples, from small numbers you know, to formulas you know, to formulas you may never have seen, to plain nonsense. Then make up your own.

A circle is every point that sits the same distance from one point in the middle. That one idea is where all the parts come from.

There are five words to know. Each one has its own color here, and keeps it in every room that follows:

• center: the point in the middle
• radius (r): from the center to the edge
• diameter (d): edge to edge, straight through the center
• circumference (C): the whole distance around the edge
• area (A): the flat space inside the edge

Tap each word above the circle to light up that part. Drag the teal dot on the edge and watch the radius swing around. It is the same length wherever it points.

A diameter is two radii laid end to end through the center. So it is always twice as long:

That is a three-number family, just like the beads. If you know one of the two lengths, you know the other.

Here is the most important fact about circles, and you can watch it happen.

Take a wheel. Put a mark on its edge, right where it touches the ground. Roll the wheel along the ground exactly one turn, until the mark comes back down to the ground.

The wheel leaves a track behind it. That track is exactly as long as the edge of the wheel, because every bit of the edge touched the ground once. The track is the circumference, laid out straight.

Now take the wheel’s diameter, the straight line across it, and lay it down along the track, again and again.

It fits three times. And there is a little bit of track left over. The little bit is about 0.14 of a diameter, about one seventh.

Try a small wheel and a big one. The track gets longer or shorter. The count does not change.

Then lay down radii instead. A radius is half a diameter, so twice as many fit: a little more than 6.28. That is where the formula C = 2 × π × r comes from.

π is not a length. It is a comparison between two lengths: a ratio.

It says: for every 1 across, a little more than 3.14 around. We say it as “3.14 to 1.”

This is true of every circle there is, large or small: a quarter, a bagel, a pizza pie, a bicycle wheel, the round park, the equator of the whole Earth.

That is not luck, and it is not something people keep rediscovering by chance. It is what π is. The number π is defined as the circumference divided by the diameter, for any circle.

Slide the size of the circle. The circle grows and its unrolled edge grows with it. The diameter still fits along the edge 3 times and a little more.

Other ways to write about 3.14

The true value of π never ends and never repeats: 3.14159265… For the GED, 3.14 is the one to use, or the π key on the calculator.

Now the two halves of this app meet.

We know that for every circle, Cd = π. That is a three-number family, just like 63 = 2. Turned into a multiplication, it reads C = π × d: the factors are π and d, and the circumference is their product. The product goes on top of the triangle.

The first one finds the distance around when you know the distance across. The second finds the distance across when you know the distance around. The third is where we started: the definition of π.

When you are given the radius

The formula sheet also says C = 2πr. That is the same formula. Two radii make a diameter, so 2 × r is just d written another way.

The plainest path: if you are given the radius, double it first to get the diameter. If you are asked for the radius, find the diameter, then halve it.

Area is the space inside, counted in squares: square inches, square feet.

The formula is A = π × r × r, written A = πr². The area is the product; π and r² are its factors. Here is where it comes from.

Cut a pizza pie into slices. Lay the slices in a row, every other one upside down, so they fit together like teeth. The more slices you cut, the more the row looks like a rectangle.

• How tall is the rectangle? One slice long: the radius.
• How wide? Half of the crust runs along the top, and half along the bottom. So the width is half the circumference, which is π × r.

The area of a rectangle is width times height:

A = πr × r = πr²

Another way to see 3.14

Draw a square on the radius: r by r. Four of those squares make a big square that just holds the circle.

The circle does not fill the big square. The corners are left out. It covers a little more than three of the four small squares: 3.14 of them.

So the area is 3.14 radius-squares, just as the circumference is 3.14 diameters. The same number, π, shows up both times.

A pizza question to argue about

Which is more pizza: one 18-inch pie, or two 12-inch pies?

One 18-inch pie: the radius is 9. A = 3.14 × 9 × 9 = 254.34 square inches.

One 12-inch pie: the radius is 6. A = 3.14 × 6 × 6 = 113.04 square inches. Two of them: 226.08 square inches.

The one big pie is more pizza, by about 28 square inches. Twelve plus twelve is more than eighteen, but area does not add up the way lengths do. Squaring the radius makes big circles grow fast.