1.4 Circles


Illustrated is a circle:

The half-life in the middle is called the radius (r). The line spanning the full distance of the circle is called the diameter (d), and is related to the radius by the formula d=2r.

The perimeter of the circle (darker purple outline of the circle) is the circumference. The circumference is approximately 3 times longer than the diameter. The exact number is 3.14159265… which is abbreviated with \pi. The relationship is thus C=\pi d, or alternatively, C=2\pi r.

Area of a circle

Area of a circle is defined as A=\pi r^2.

For example, if a circle has a radius of r=16cm, rounding off to 2dp, the answer is A=\pi r^2=\pi 16^2=\pi*256=804.25cm^2.

Area of a donut

A donut is analogous to a circle, but lacks a centerpiece.

When in a spherical shape, the donut is known as a torus.

Given that the radius of the larger circle is 10m and the area of the smaller circle is 4m, the area of the donut can be found as the area of the larger circle minus the area of the smaller circle. Alternatively, this is A=(\pi r^2)-(\pi r^2)=(\pi 10^2)-(\pi 4^2)=264m^2.

Area of a curved rectangle

Given that the radius is r=6m and x=23m, the area of the curved rectangle is actually equal to the area of a whole circle, and the rectangle created by x*2r. This equation is otherwise, \dfrac{1}{2}\pi r^2+2r*x+\dfrac{1}{2}\pi r^2=\dfrac{1}{2}\pi (6)^2+2(6)*(23)+\dfrac{1}{2}\pi (6)^2=390m^2.

Area of a shaded area out of a circle

Given that r=4m, the area of the equivalent square is r^2. The area of the circle in that region is \dfrac{1}{4}\pi r^2. The area of the shaded area is thus r^2-\dfrac{1}{4}\pi r^2=r^2-\dfrac{1}{4}*\pi*4^2=3.43m^2.


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