The circle is one of the most important shapes in math. Let’s take a look at some key terms that are used to measure and define circles.
First, a chord is a line that connects any point on a circle with another point on the same circle.
This is a chord
Here’s another chord
And here’s another
The diameter of a circle is arguably its most important chord. The diameter of a circle divides it perfectly in half; it is a chord that crosses the circle through its center.
Because the diameter passes through the center, it is also the longest possible chord for a given circle.
The radius is defined as a line that connects the center of a circle with a point on the circle.
Because a circle’s radius touches the center, it is actually one half of the diameter of the circle.
In other words, two radii - that’s the plural for radius - add up to one diameter.
We can also express this with an equation: d = 2r, or r = ½d, where d refers to diameter and r refers to radius.
Part of why the diameter and radius of a circle are important measurements is because we can use them to calculate a circle’s circumference and area. Let’s take a look at circumference first.
A circle’s circumference is the measure of the length of the outside of the circle, all the way around.
In other shapes, like rectangles and triangles, this is the measurement you’d refer to as the “perimeter.” So, “circumference” is basically a special word for the perimeter of a circle.
How do we calculate the circumference of a circle? That’s where a special number known as pi comes in. It is represented by a symbol that looks like this.
Pi, which is typically approximated as 3.14, is the ratio between any circle’s circumference and diameter. That is, for any circle you can imagine, the circumference, or distance around the circle, is approximately 3.14 times larger than the diameter of the circle. Pi is an irrational number whose decimal values go on forever - its digits start off with 3.14159 and go on from there - so we typically just use 3.14 as a close approximation.
This ratio allows us to calculate the circumference of any circle we know the diameter of, using a simple formula. C, the circumference, equals pi times d, the diameter.
Remember how the diameter of a circle is equal to two times the radius? For that reason, we can also write this formula as C = 2𝜋r; C, the circumference, is equal to 2 times pi times the radius
Let’s look at an example: here’s a circle with a radius of 3 cm. What is its circumference?
Since we know that our radius is 3 cm, we can use the equation C = 2𝜋r. If we substitute 3 for r, we find that C equals 2 times pi times 3, or 6. Some math problems will ask for the answer in terms of pi, and in that case, we’d be done here - the circumference is 6 cm.
Other times, you may be asked to use 3.14 for pi, in which case our answer would be 6 times 3.14, or 18.84 cm.
On tests where a calculator is allowed, you may need to use the pi button on the calculator and then round your answer to a particular digit. If we did that for 6𝜋, we’d get this number. If we were asked to round that to the hundredths place, we’d get 18.85 cm.
On standardized tests, you will also need to make sure you know whether you’ve been given the radius or the diameter of the circle - if you’re given the radius but use it as the diameter or vice versa, you’ll get the wrong answer every time!
In this case, our diameter would be 6 - which is two times our radius of 3. Using the formula C = 𝜋d would then give us the same answer: C = 𝜋 times 6, or 6𝜋 cm.
If you remember these formulas and read each question carefully, you can get circumference problems right every time!
Let’s shift gears now and talk about area. Area is a measure of the space enclosed by a shape, and you’re likely very familiar with finding the area of polygons like rectangles and triangles. For circles, we just have one more formula to remember: area, or A, is equal to pi times the radius squared.
Looking back at our circle from before, with a radius of 3 cm, we can substitute 3 for r in and find that the area of this circle would be pi times 3 squared. Since 3 squared is 9, our area is 9𝜋 cm2.
Again, you may need to use 3.14 for pi, or plug the numbers into your calculator and give a rounded answer, If we use 3.14 for pi, we would get 28.26 cm2.
Sometimes, you will be asked to find the length of an arc on a circle, or the area of a sector of a circle. In both cases, we’re just being asked to calculate the area or circumference of one slice of a complete circle, and we can use the skills we’ve already learned to do so.
Let’s say that we need to find the area of this slice of the circle we looked at before.
We’ll usually be given the central angle measurement for the slice of the circle we’re being asked to measure. In this case, the central angle is 120°.
Let’s think for a moment about what fraction of the circle this slice takes up. There are 360° in a whole circle, and this slice takes up 120°. So, we can represent the size of the slice as a fraction of the circle as a whole: 120 out of 360. If we reduce this fraction, we get ⅓. In other words, this slice is ⅓ of the size of the whole circle.
Now, all we have to do to find the area of this sector is multiply the area of the whole circle by ⅓. We calculated earlier that the area of a circle with a radius of 3 cm is 9𝜋 cm2, using the formula A = 𝜋r2, so our sector area is ⅓ of that: 3𝜋 cm2.
If we’re asked to find the arc length for this slice, we’re looking at a fraction of the whole circle’s circumference. The circumference of the circle was 6𝜋 cm, which we got using the formula C = 2𝜋r, and we know that this arc is ⅓ the size of the circle’s entire circumference. So, our arc length is ⅓ of 6𝜋 - or, 2𝜋 cm.
We can sum up sector area and arc length with these formulas:
Where theta is equal to the measure of the central angle of the slice of the circle that we’re looking at, in degrees.