Understanding the Area and Circumference of a Circle

Factoid: The area of a circle is pi (approximately 3.14) times the radius of the circle squared. The circumference is pi times the diameter.

You are here probably because you do not remember these formulas. Your brain was not built to remember meaningless formulas. Keep reading to understand these formulas (or how to learn math).

Area of a Circle

Draw a circle. Now draw a square just surrounding the circle. Now think about the area of the circle in relationship to the area of the square. Obviously, the area of the cirle is less. But how much less? In this drawing, the area of the cirle is about 78.5% of the area of the square.

The key things is, that percentage is the same no matter what the size of the circle. So, you can always draw a square around a circle, calculate the area of the square, and then the area of the circle is 78.5% of that.

Okay, let's do a formula. The line from one side of the circle to the other is called the diameter. It is usually abreviated as d. So:

  • 1. The length of the side of the square is also d
  • 2. The area of the square is d squared (d times d), and

3. The area of the circle is .785 times d squared

Wasn't that easy? (relationship to traditional formula)


The circumference of the circle -- the line around the outside of the circle -- can be calculated in the same way. The circumference of the circle is just a percentage of the circumference (perimeter) of the square, and it's the same percentage no matter what the size.

Believe it or not, that percentage is 78.5% again. (Isn't that incredible? A seemingly random number is the answer to two different problems.)

  • The sides of the square are d
  • the perimeter of the square is 4 * d (because there are four sides)

So the formula for the circumference of a circle is .785 * 4 * d.

Mathematicians and Pi

Maybe you would want to give this ratio, which is approximately 78.5%, a name. I will call it cake. (I used to call it pie.) Then the formula for the area of a circle is exactly cake times d squared.

Um, if your teacher wants an exact answer, don't use cake, because that is a word I made up. (Mnemonic: Don't leave cake on your homework.)

Fortunately, there is a well known word, called pi. Pi is 4 times cake, or approximately 3.14. You can use pi on your homework, you just have to change the formulas.
area = d * d * cake
area = d * d * pi / 4
area = r * r * pi
circumference = 4 * d * cake
circumference = d * pi

The letter 'r' in the third formula stands for the radius of the circle. The radius is half of the diameter.

If you are willing to put the radius in one formula and the area in the other formula, then you can express area and circumference more simply using pi. So, as long as you do not care whether people understand (or remember) the formulas, pi is a slightly better choice than cake. (Mathematicians!)

Bob Sez

If you ever need to calculate the area or circumference of a circle, first draw the surrounding square. Second, calculate the area or circumference (perimeter) of the square. Then multiply by cake. That's about 78.5%, if you need a number. If you need to be exact, substitute pi/4 for cake.

This website is about teaching math through insight, which teaches mental models, understand, and concepts. Not memorization (which you forget).