Area of a Circle

Area of a circle may seem complicated when all you’ve done throughout your entire math career is find the area of shapes with straight lines. 

Most area formulas are simple.  They all involved base and height in some form or another.  Just when you think you’ve got the hang of this area thing a circle comes along.

The formula is: $$A=\pi r^2$$

Where is the base?!  Where is the height!?  This is unlike any area formula you’ve ever seen before. 

The area of a circle formula may throw your brain for a loop (pun intended) the first time you see it.  But fear not!  It may be different, but it’s not difficult.  You can handle it.

Vocabulary

Before we tackle any area of a circle problems let’s review the vocabulary we need to know.

•  $$\pi$$ is the simple for pi.  Pi is an irrational number that is often rounded to 3.14.
•  $$r$$ is the radius of a circle.  The radius is the distance from the center to the edge of a circle.  Be sure not to confuse radius with diameter (which is the distance across a circle when passing through the center).

Simple Practice

Now that you’ve got the vocabulary down let’s tackle a simple problem using the formula for area of a circle.

Find the area of the circle given a radius of 4 cm.

Remember, the formula for area of a circle is $$A=\pi r^2$$

All we have to do is substitute in 4 cm for the radius.

We’ll end up with this equation:   $$A=\pi{(4)}^2$$

Which simplifies to $$A=16\pi$$

If we estimate pi as 3.14 and solve we’ll get $$A=50.24cm^2$$

The area of this circle is 50.24 square centimeters.  Great job!  First problem down.

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Decimal Practice

Now that you’ve got one practice problem under your belt let’s try another.  This time, we’ll ramp up the difficulty by involving decimals.  Remember, even though it may seem complicated all we have to do is apply the formula.  You’ve got this!

Find the area of the circle given a radius of 36.8 cm.

Remember, the formula for area of a circle is $$A=\pi r^2$$

All we have to do is substitute in 36.8 m for the radius.

We’ll end up with this equation: $$A=\pi{(36.8)}^2$$

(Wow! Squaring a larger number really makes a difference!)

Which simplifies to $$A=1,354.24\pi$$

If we estimate pi as 3.14 and solve we’ll get $$A=4,252.3136m^2$$

The area of this circle is $$4,252.3136$$ square meters.  Nicely done.  Don’t let the big numbers scare you!  The process is the same.

What if you’re not given the radius?!

Just when you think you’ve got it down and could use the $$A=\pi r^2$$  formula in your sleep you come across a problem like this one.

Find the area of the circle given a diameter of 51 in.

Wait a minute.  To solve for the area of a circle you need the radius.  This problem doesn’t tell you the radius.  What should you do?

It’s really not as hard as it seems.  Remember way back to the beginning when we reviewed the vocabulary?

We looked at this diagram.

Notice that the diameter is in blue.  It is TWICE the radius.  That means, that to

get the radius (which is what we need to solve our problem) is simply divide the diameter in half.

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The problem tells us that the diameter is 51 in.

 $$\frac{51}2$$ is $$25.5$$ so $$25.5$$ is our radius $$r=25.5 in$$.

Now that we’ve figured out the radius we’re ready to solve!

Remember, the formula for area of a circle is $$A=\pi r^2$$

All we have to do is substitute in 25.5 in for the radius.

We’ll end up with this equation:   $$A=\pi{(25.5)}^2$$

Which simplifies to $$A=650.25\pi$$

If we estimate pi as 3.14 and solve we’ll get $$A=2,041.785 in^2$$

Now we know that the area of a circle with a diameter of 51 in is 2,041.785 square inches.  Problem solved!

Real World Problems

Now that you know how to tackle area of a circle problems.  Let’s take a look at some real world situations where your new knowledge might come in handy.

You need to find a wheel that’s the right size for your tire.

The diameter of the inside of your tire measures 24 inches.

What is the area of the wheel that will fit your tire properly?

First, you need to find the radius.  If the diameter is 24 inches and diameter is twice the radius you’ll need to divide by two.

$$\frac{24}2=12$$.  So $$12=r$$

Now that you know the radius let’s substitute that new information into the equation.

When we substitute, we’ll get equation: $$A=\pi{(12)}^2$$

Which simplifies to $$A=144\pi$$

If we estimate pi as 3.14 and solve we’ll get $$A=452.16 in^2$$

You’ll need to choose the wheel with an area of 452.16 square inches.

Now let’s consider another problem.

You need to find a circular rug for your front hall. The diameter must be 48 inches for the rug to properly fill the space. 

How large of a rug should you buy?

Once again, we are not given the radius.  Finding it will be our first step in solving the problem.  If the diameter (twice the radius) is 48 inches, we’ll simply need to divide by two.

$$\frac{48}2=24$$ inches so $$24=r$$.

Now, using that new information we can solve for A.

When we substitute, we’ll get equation: $$A=\pi{(24)}^2$$

Which simplifies to $$A=576\pi$$

If we estimate pi as 3.14 and solve we’ll get $$A=1.808.64 in^2$$

You’ll need to find a rug with an area of 1,808.64 square inches.

Wrapping Up

You’ve mastered it!  The area of a circle may seem odd at first, but once you know the formula you are good to go.  Simply substitute in the radius and solve away.  Well done.