Circle

What Is a Circle?

A circle is the set of all points in a plane that are at a fixed distance from a specific point, called the center. Imagine pinning down a piece of string at one end and tying a pencil to the other. If you pull the string taut and draw, keeping the pinned end fixed, the shape you create is a perfect circle. Every point on the line you drew is the exact same distance from the pin.

To understand circles, we need to learn their vocabulary. These terms describe the different parts of a circle and the lines associated with it:

• Center: The point inside the circle that is equidistant from all points on the circle itself. We often label it with a letter, like point O.
• Radius (r): A line segment from the center of the circle to any point on the circle. It represents the fixed distance in the definition of a circle.
• Diameter (d): A line segment that passes through the center of the circle and has both of its endpoints on the circle. The diameter is always twice the length of the radius ($d=2r$).
• Chord: A line segment whose two endpoints lie on the circle. The diameter is the longest possible chord in a circle.
• Circumference (C): The total distance around the outside of the circle. It's like the perimeter of a polygon, but for a curved shape.
• Arc: A portion of the circumference of a circle. Think of it as the crust of a single slice of pizza.
• Tangent: A line that touches the circle at exactly one point, called the point of tangency. It skims the edge without ever crossing inside.
• Secant: A line that passes through a circle, intersecting it at two distinct points.

Understanding these parts is the first step to mastering all the calculations you can perform with circles.

What Is the Magical Number Pi (π)?

You can't study circles without meeting one of the most famous numbers in all of mathematics: Pi, represented by the Greek letter $\pi$. Pi is a special, constant value that expresses the relationship between a circle's circumference and its diameter.

Specifically, Pi is the ratio of the circumference of any circle to its diameter. This means if you take any circle, no matter how big or small, and divide its circumference by its diameter, you will always get the same number: $\pi$.

Pi is an irrational number. This means its decimal representation goes on forever without ever repeating in a pattern. The first few digits are $3.1415926535...$, but for most calculations in middle school, you will use one of two common approximations:

• Decimal Approximation: $\pi \approx 3.14$
• Fractional Approximation: $\pi \approx \frac{22}{7}$

Your teacher or textbook will usually tell you which approximation to use. The fractional one is often handy when the radius or diameter is a multiple of $7$. Because $\pi$ is essential for calculating both the distance around a circle and the space inside it, it's a number you'll use constantly when solving circle problems.

How Do You Calculate a Circle's Circumference?

The circumference is the distance around a circle. Since we know that $\pi$ is the ratio of circumference to diameter ($C/d$), we can rearrange that relationship to create a formula for finding the circumference.

There are two primary formulas for calculating circumference, and which one you use depends on whether you are given the radius or the diameter.

1. Using the Diameter (d):

2. Using the Radius (r): Since the diameter is twice the radius ($d=2r$), we can substitute $2r$ for $d$ in the first formula to get:

Both formulas give you the exact same result. Let's work through some examples.

How Do You Find the Area of a Circle?

While circumference measures the distance around a circle, area measures the total space inside it. The formula for the area of a circle also uses $\pi$, but it involves the radius squared.

It is very important to remember the order of operations here: you must square the radius first, and then multiply the result by $\pi$. A common mistake is to multiply the radius by $2$ instead of squaring it, which would give you the circumference formula. Area is always measured in square units, such as cm${}^2$, m${}^2$, or in${}^2$.

What if you're only given the diameter? You must first find the radius by dividing the diameter by $2$ ($r=d/2$) before you can use the area formula.

What Are Arcs and Sectors?

Sometimes you don't need to measure the entire circle, but just a piece of it. This is where arcs and sectors come in. Both are defined by a central angle, which is an angle whose vertex is the center of the circle.

• An arc is a part of the circumference. The length of an arc is a fraction of the total circumference.
• A sector is a region bounded by two radii and the arc between them. It looks like a slice of pizza. The area of a sector is a fraction of the total area.

The fraction for both calculations is determined by the central angle, which we'll call $\theta$ (theta), measured in degrees. Since a full circle has ${360}^{\circ }$, the fraction is $\frac{\theta }{360}$.

Formula for Arc Length:

Formula for Sector Area:

Comparing Key Circle Terminology

It can be easy to mix up the different lines and segments associated with a circle. This table provides a quick comparison of the most important terms.

Common Mistakes to Avoid with Circles

When working with circles, a few common errors can trip students up. Being aware of them is the best way to avoid making them!

• Confusing Radius and Diameter: This is the most common mistake. Always double-check if a problem gives you the radius or the diameter. If you're given the diameter, you must divide it by two before using the area formula $A=\pi r^2$.
• Mixing Up Area and Circumference Formulas: The formulas $A=\pi r^2$ and $C=2\pi r$ look similar. Remember that area is always measured in square units, so its formula has the radius squared ($r^2$). Circumference is a length, so its formula is not squared.
• Forgetting to Square the Radius: For area, students sometimes multiply the radius by $2$ instead of squaring it (multiplying it by itself). Remember, $r^2$ means $r\times r$, not $2\times r$. For a radius of $5$, $r^2$ is $25$, not $10$.
• Rounding Too Early: When a problem requires a decimal answer, use the $\pi$ button on your calculator or a more precise value like $3.14159$ for intermediate steps. Only round your final answer to the specified number of decimal places to maintain accuracy.
• Incorrect Units: Remember to include the correct units in your answer. Circumference is a length (e.g., cm, in, ft), while area is measured in square units (e.g., cm${}^2$, in${}^2$, ft${}^2$).

Quick Summary and Formula Reference

Here is a quick reference for the most important concepts and formulas related to circles. Use this as a study guide to refresh your memory.

• Radius (r): Distance from the center to the edge.
• Diameter (d): Distance across the circle through the center.
• Relationship: $d=2r$ and $r=d/2$

Key Formulas:

• Circumference (using radius):
  $C=2\pi r$
• Circumference (using diameter):
  $C=\pi d$
• Area:
  $A=\pi r^2$

Always remember to check whether you have the radius or diameter and to use the correct units in your final answer (units for length, square units for area).