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Reference

Concept

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 $C.$

circle $O,$since it is centered at $O.$

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

Rule

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

$C=πd$

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

$C=2πr$

Consider two circles and their respective diameters and circumferences.

By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional.$C_{A}C_{B} =d_{A}d_{B} $

This proportion can be rearranged.
$C_{A}C_{B} =d_{A}d_{B} $

▼

Rearrange equation

MultEqn

$LHS⋅C_{A}=RHS⋅C_{A}$

$C_{B}=d_{A}d_{B} (C_{A})$

DivEqn

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

$d_{B}C_{B} =d_{A}1 (C_{A})$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$d_{B}C_{B} =d_{A}C_{A} $

$dC =π⇒C=πd$

Rule

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

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 $2πr,$ the arc length of each semicircle is half this value, $πr.$

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.

Here, the shorter sides become

$A=πr⋅r⇔A=πr_{2}$

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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