Circle
A circle is a round, two-dimensional shape. All points on the edge of the circle are at the same distance from the center.
The radius of a circle is a line from the center of the circle to a point on the side. Mathematicians use the letter <math>r</math> for the length of a circle's radius. The center of a circle is the point in the very middle. It is often written as <math>O</math>.
The diameter (meaning "all the way across") of a circle is a straight line that goes from one side to the opposite and right through the center of the circle. Mathematicians use the letter <math>d</math> for the length of this line. The diameter of a circle is equal to twice its radius (<math>d</math> equals <math>2</math> times <math>r</math>):[1]
- <math>d=2r</math>
The circumference (meaning "all the way around") of a circle is the line that goes around the center of the circle. Mathematicians use the letter <math>c</math> for the length of this line.[2]
The number <math>\pi</math> (written as the Greek letter pi) is a very useful number. It is the length of the circumference divided by the length of the diameter (<math>\pi</math> equals <math>c</math> divided by <math>d</math>). As a fraction the number <math>\pi</math> is equal to about <math>\frac{22}{7}</math> or <math>\frac{355}{113}</math> (which is closer) and as a number it is about <math>3.1415926536</math>.
| <math>\pi=\frac cd</math> | |
| <math>\therefore\textrm{(therefore)}</math> | <math>c=2\pi r</math> |
| <math>c=\pi d</math> |
The area, <math>A</math>, inside a circle is equal to the radius multiplied by itself, then multiplied by <math>\pi</math> (<math>A</math> equals <math>\pi</math> times <math>r</math> times <math>r</math>).
- <math>A=\pi r^2</math>
Calculating π[change]
<math>\pi</math> can be measured by drawing a circle, then measuring its diameter (<math>d</math>) and circumference (<math>c</math>). This is because the circumference of a circle is always equal to <math>\pi</math> times its diameter.[1]
- <math>\pi=\frac cd</math>
<math>\pi</math> can also be calculated by only using mathematical methods. Most methods used for calculating the value of <math>\pi</math> have desirable mathematical properties. However, they are hard to understand without knowing trigonometry and calculus. However, some methods are quite simple, such as this form of the Gregory-Leibniz series:
- <math>\pi=\frac41-\frac43+\frac45-\frac47+\frac49-\frac{4}{11}+\cdots</math>
While that series is easy to write and calculate, it is not easy to see why it equals <math>\pi</math>. A much easier way to approach is to draw an imaginary circle of radius <math>r</math> centered at the origin. Then any point <math>(x,y)</math> whose distance <math>d</math> from the origin is less than <math>r</math>, calculated by the Pythagorean theorem, will be inside the circle:
- <math>d=\sqrt{x^2+y^2}</math>
Finding a set of points inside the circle allows the circle's area <math>A</math> to be estimated, for example, by using integer coordinates for a big <math>r</math>. Since the area <math>A</math> of a circle is <math>\pi</math> times the radius squared, <math>\pi</math> can be approximated by using the following formula:
- <math>\pi=\frac{A}{r^2}</math>
Calculating measures of a circle[change]
Area[change]
Using the radius: <math>A=\pi r^2=\frac{\tau r^2}{2}</math>
Using the diameter: <math>A=\frac{\pi d^2}{4}=\frac{\tau d^2}{8}</math>
Using the circumference: <math>A=\frac{c^2}{2\tau}=\frac{c^2}{4\pi}</math>
Circumference[change]
Using the radius: <math>c=\tau r=2\pi r</math>
Using the diameter: <math>c=\pi d=\frac{\tau d}{2}</math>
Using the area: <math>c=\sqrt{2\tau A}=2\sqrt{\pi A}</math>
Diameter[change]
Using the radius: <math>d=2r</math>
Using the circumference: <math>d=\frac c\pi=\frac{2c}{\tau}</math>
Using the area: <math>d=2\sqrt\frac A\pi=2\sqrt\frac{2A}{\tau}</math>
Radius[change]
Using the diameter: <math>r=\frac d2</math>
Using the circumference: <math>r=\frac c\tau=\frac{c}{2\pi}</math>
Using the area: <math>r=\sqrt\frac A\pi=\sqrt\frac{2A}{\tau}</math>
Related pages[change]
References[change]
- ↑ 1.0 1.1 Weisstein, Eric W. "Circle". mathworld.wolfram.com. Retrieved 2020-09-24.
- ↑ "Basic information about circles (Geometry, Circles)". Mathplanet. Retrieved 2020-09-24.
