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Reference

Circles

Concept

Circle

A circle is the set of all the points in a plane that are equidistant from a given point. There are a few particularly notable features of a circle.

Center - The given point from which all points of the circle are equidistant. Circles are often named by their center point.
Radius - A segment that connects the center and any point on the circle. Its length is usually represented algebraically by $r .$
Diameter - A segment whose endpoints are on the circle and that passes through the center. Its length is usually represented algebraically by $d .$
Circumference - The perimeter of a circle, usually represented algebraically by $C .$

The following circle can be referred to as

⊙ O,

or circle

O,

since it is centered at

O .

Parts of a circle

In any given circle, the lengths of any radius and any diameter are constant. They are called the radius and the diameter of the circle, respectively.

Rule

Circumference of a Circle

The circumference of a circle is calculated by multiplying its diameter by $π .$

$C = π d$

This can be visualized in the following diagram.

Animation unrolling a circle

Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying $2 r$ by $π .$

$C = 2 π r$

Proof

Consider two circles and their respective diameters and circumferences.

By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional.

\frac{C _{B}}{C _{A}} = \frac{d _{B}}{d _{A}}

This proportion can be rearranged.

\frac{C _{B}}{C _{A}} = \frac{d _{B}}{d _{A}}

Rearrange equation

$LHS \cdot C_{A} = RHS \cdot C_{A}$

C_{B} = \frac{d _{B}}{d _{A}} (C_{A})

$LHS / d_{B} = RHS / d_{B}$

\frac{C _{B}}{d _{B}} = \frac{1}{d _{A}} (C_{A})

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

\frac{C _{B}}{d _{B}} = \frac{C _{A}}{d _{A}}

Therefore, for any two circles, the ratio of the circumference to the diameter is always the same. This means that this ratio is constant. This constant is defined as

π .

With this information, it can be shown that the circumference of a circle is the product between its diameter and

π .

$\frac{C}{d} = π \Rightarrow C = π d$

Rule

Area of a Circle

The area of a circle is the product of $π$ and the square of its radius.

Circle

Proof

Informal Justification

A circle with radius $r$ will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is $2 π r,$ the arc length of each semicircle is half this value, $π r .$

Now, the above sectors will be unfolded. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. As such, the area of the figure below should be the same as the circle's area.

It can be noted that if the circle is divided into more and smaller sectors, then the figure will begin to look more and more like a rectangle.

Here, the shorter sides become more vertical and the longer sides become more horizontal. If the circle is divided into infinitely many sectors, the figure will become a perfect rectangle with base

π r

and height

r .

Since the area of a rectangle is the product of its

height

and its

base,

the following formula can be derived.

$A = π r \cdot r \Leftrightarrow A = π r^{2}$

It has been shown that the area of a circle is the product of $π$ and the square of its radius.

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