Using the Pythagorean Theorem with Circles

The Pythagorean Theorem can help us find the radius of a circle when we know its circumference. To do this, we need to understand how the radius, diameter, and circumference are connected.

What You Need to Know

First, let's look at the formula for the circumference of a circle. The circumference ($C$) can be calculated using this formula:

$$C = 2\pi r$$

In this formula, $r$ stands for the radius of the circle.

Finding the Radius

If we want to find the radius using the circumference, we can change the formula a bit:

$$r = \frac{C}{2\pi}$$

Let’s say the circumference of a circle is 31.4 units. To find the radius, we can do this:

$$r = \frac{31.4}{2\pi} \approx 5$$

So, the radius is about 5 units.

Linking to the Pythagorean Theorem

Now, let's think about a circle that fits inside a right triangle. The Pythagorean Theorem can help us relate the radius of this circle to the triangle's sides.

In a triangle with two sides called $a$ and $b$, and the longest side is $c$ (called the hypotenuse), the radius $r$ of the circle inside the triangle can be found using this formula:

$$r = \frac{A}{s}$$

Here, $A$ is the area of the triangle, and $s$ stands for the semi-perimeter (which is half of the triangle's perimeter).

We can also find the triangle's area using the Pythagorean Theorem, which helps us understand more about the properties of the circle.

So, the Pythagorean Theorem isn’t just for triangles—it can also help us learn about circles!