The radius will generally be provided to you. It can be difficult to measure to the exact center of a circle, unless the center is already marked for you on a circle drawn on paper. For this example, assume that you are told that the radius of a given circle is 6 cm.
Do not get confused and square the entire equation. For the sample circle with radius, r=6{\displaystyle r=6}, then r2=36{\displaystyle r^{2}=36}.
For the given example with a radius of 6 cm, the area is calculated as: A=πr2{\displaystyle A=\pi r^{2}} A=π62{\displaystyle A=\pi 6^{2}} A=36π{\displaystyle A=36\pi } or A=36(3. 14)=113. 04{\displaystyle A=36(3. 14)=113. 04}
For the sample circle with a radius of 6 cm, the area will be either 36π{\displaystyle \pi } cm2 or 113. 04 cm2.
Assume for this example that the diameter of your circle is 20 inches.
Therefore, the sample circle with a diameter of 20 inches will have a radius of 20/2, or 10 inches.
A=πr2{\displaystyle A=\pi r^{2}} A=π102{\displaystyle A=\pi 10^{2}} A=100π{\displaystyle A=100\pi }
You can also provide the numerical approximation by multiplying by 3. 14 instead of π{\displaystyle \pi }. This will give a result of (100)(3. 14) = 314 sq. in.
A=C24π{\displaystyle A={\frac {C^{2}}{4\pi }}}
For this example, assume that you have been told or have measured that the circumference of a circle (or circular object) is 42 cm.
C=π2r{\displaystyle C=\pi 2r} C2π=r{\displaystyle {\frac {C}{2\pi }}=r}…. . (divide both sides by 2π{\displaystyle \pi })
A=πr2{\displaystyle A=\pi r^{2}}…. . (original area formula) A=π(C2π)2{\displaystyle A=\pi ({\frac {C}{2\pi }})^{2}}…. . (substitute equality for r) A=π(C24π2){\displaystyle A=\pi ({\frac {C^{2}}{4\pi ^{2}}})}…. . (square the fraction) A=C24π{\displaystyle A={\frac {C^{2}}{4\pi }}}…. . (cancel π{\displaystyle \pi } in numerator and denominator)
For this sample, you were given C=42{\displaystyle C=42} inches. A=C24π{\displaystyle A={\frac {C^{2}}{4\pi }}} A=4224π{\displaystyle A={\frac {42^{2}}{4\pi }}}…. . (insert value) A=17644π{\displaystyle A={\frac {1764}{4\pi }}}. …. (calculate 422) A=441π{\displaystyle A={\frac {441}{\pi }}}…. . (divide by 4)
For this sample circle, with a circumference given as 42 cm, the area is 441π{\displaystyle {\frac {441}{\pi }}} sq. cm. If you approximate, 441π=4413. 14=140. 4{\displaystyle {\frac {441}{\pi }}={\frac {441}{3. 14}}=140. 4}. The area is approximately equal to 140 sq. cm.
Make sure you know if you are measuring the small angle between the two radii or the greater angle outside them. The problem you are working on should define this for you. The sum of the small angle and the great angle will be 360 degrees. In some problems, instead of having you measure the central angle, the problem may just tell you the measurement. For example, you might be told, “The central angle of the sector is 45 degrees” or you may be expected to measure it.
Acir=Asec360C{\displaystyle A_{cir}=A_{sec}{\frac {360}{C}}} Acir{\displaystyle A_{cir}} is the area of the full circle Asec{\displaystyle A_{sec}} is the area of the sector C{\displaystyle C} is the central angle measure
Acir=Asec360C{\displaystyle A_{cir}=A_{sec}{\frac {360}{C}}} Acir=15π36045{\displaystyle A_{cir}=15\pi {\frac {360}{45}}} Acir=15π(8){\displaystyle A_{cir}=15\pi (8)} Acir=120π{\displaystyle A_{cir}=120\pi }
If you want to report a numerical value, you can multiply 120 x 3. 14 to get a value of 376. 8 cm2.
