# Circles and their properties

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## Introduction to Circles and their Properties

One of the most interesting and unique shapes we study in geometry is a circle. It has no straight lines, no angles, and no vertices. A circle is simply a perfectly round shape. Each point of this line is at the same distance from a point inside the circle, which is referred to as the center of the circle. In our daily life, we see that a pizza resembles a circle, a cake is a circle, a bangle is a circle, the wheels of a bicycle are circles, a Ferris wheel is a circle and so many more.

## The Big Idea: What is Pi?

Before any discussion of a circle, a very interesting number needs to be understood. As it turns out, the ratio of the circumference to the diameter of any circle is always the same. It is referred to as pi and is also represented as $${π}$$

Pi mathematically is the ratio of the circumference of the circle to its diameter and is a constant for all circles. Take ANY circle of ANY size, and the value of this ratio will remain unchanged!

Ancient Babylonians calculated this value an used an approximation of 3. Later ancient Egyptians refined the calculations and found it to be 3.106. Today, we know Pi as:

It is a non-terminating, non-repeating number. For ease of use, mathematically we approximate it to 3.14 or $${22 \over 7}$$.

### Parts of a circle

Here are the brief definitions of the parts of a circle:

• Radius: The distance from the center of a circle to any point on its edge
• Circumference: The total length of the rim which bounds a circle
• Chord: A line that joins any two points on the circumference of the circle
• Secant: A chord that is extended beyond the two points where it meets the circle
• Diameter: A chord that passes through the center of a circle. It is always two times the radius
• Tangent: A straight line that touches the circle's circumference at only one point
• Tangent circles: Two circles which touch each other at just one point
• Concentric Circles: Two circles with the same center but different diameter (and therefore different radius and circumference)
• Congruent Circles: Just like we discussed the congruency of triangles, we can call two circles congruent if they have different centers but exactly same radiu

### Area and circumference of a circle

We discussed pi and its value above. Based on the constant pi, we have the formulas for the circumference and area of a circle.

$${\text{Area}=πr^2}$$

$${\text{Circumference}=2πr}$$

In both the above formulae, the letter $${r}$$ stands for the radius of the circle. Since the diameter of a circle is twice the radius, therefore in the formula $${\text{Circumference}=2πr}$$, we can replace $${2r}$$ by the diameter D and write the formula as $${\text{circumference}=πD}$$, which was the original way of describing or calculating the constant $${π}$$.

### Properties of a circle

Based on the different parts of a circle mentioned above, there are several interesting and important properties of a circle. Here is a list of 12 of the most important properties of a circle:

1. Two circles having equal radius are congruent to each other and also similar to each other.
2. The central angle which intercepts an arc is the double of any inscribed angle that intercepts the same arc.
3. The radius perpendicular to a chord bisects the chord.
4. The chords equidistant from the center are equal in length.
5. A tangent to a circle is at a right angle to the radius at the point of contact.
6. Two tangents drawn on a circle from a point outside are equal in length.
7. The angle subtended at the center of a circle by its circumference is equal to four right angles.
8. The circumference of two different circles is proportional to their corresponding radii.
9. Arcs of the same circle are proportional to their corresponding angles.
10. Radii of the same circle or equal circles are equal.
11. The diameter of a circle is the longest chord

### Some interesting facts about circles

Did you know that for a given perimeter (circumference), the circle is the shape which covers the maximum area? Conversely, for a given area, a circle has the lowest perimeter. Also, have you noticed that an animal or fish represent many signs of the zodiac? The word zodiac is derived from the Greek word kyklos (which means circle) and zoon (which means animal), which gives the meaning ‘circle of animals’.