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Formulas for Circles and Spheres

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In geometry, a circle is a 2-dimensional shape such that every point on the outside has the same distance to the center. A sphere is a 3-dimensional version of a circle. The distance from the center to the outside of a circle or sphere is called its radius. To calculate the circumference, area, surface area, or volume, all you need know is the radius of the circle or sphere, and the appropriate formula. These math equations are described below.

Notice that in each formula below, the number π ≈ 3.14159... appears. This is a number frequently encountered in many areas of math, especially in geometry and trigonometry. It was first discovered in conjunction with the study of circles. Formally, π is defined to be the ratio of a circle's circumference to its radius. Although π is an irrational number (a non-repeating decimal), you can obtain accurate results using the estimation 3.14159 or 3.14.

Circumference of a Circle

To find the perimeter, aka circumference of a circle, use the formula C = 2 πr. For example, if a circle has a 5 cm radius, its circumference is C = 2(3.14)(5) = 31.4 cm.

Area of a Circle

To find the amount of 2-dimensional space a circle occupies, i.e. its area, use the formula A = πr². For example, if a circle has an 8 cm radius, its area is A = (3.14)(64) = 200.96 cm².

Surface Area of a Sphere

Surface area is the amount of 2-dimensional space on the outside of a 3-dimensional geometric figure. For a sphere, the surface area equation is SA = 4πr². For instance, suppose a sphere has a 10 cm radius. Then its surface area is SA = 4(3.14)(100) = 1256 cm².

Volume of a Sphere

Volume is the amount of 3-dimensional space inside a sphere. The formula for volume is V = (4/3)πr³. For example, a sphere whose radius is 9 cm has a volume V = (4/3)(3.14)(729) = 3052.08 cm³.

Additional Notes

There are many applications for these geometry formulas. To find the volume of a cylindrical container, for instance, you just multiply the area of the base (which is a circle) and the height of the container. The formula for cylindrical volume is πr²h. Or to find the amount of air inside an inflated ball, just use the spherical volume formula.

If you have studied calculus before, you may notice that the equation for circumference is the derivative of the equation for area, and the surface area formula is the derivative of the volume formula. You may also notice that when the formula measures 3-dimensional space, the exponent of r is 3; and when the formula measures 2-dimensional space, the exponent of r is 2.