A sphere is a three-dimensional circle. Or you could say that a sphere is the set of all the points that are at the same distance from the center of the sphere. In nature, centrifugal force and gravity tend to make a lot of things into spheres: soap bubbles, for instance, atoms, and planets.
The radius of a sphere is the distance from its center to any point on its surface. The surface area of a sphere is the set of all the points on the outside of the sphere. To figure out what the surface area of a sphere is, you multiply the radius by itself and then multiply that by pi, so the formula is 4πr2. This is because the area is the length times the width (just like the area of a square). The width of a sphere is its diameter (twice the radius, or 2r). The length of a sphere is its circumference (2πr). So the width times the length, or the area, is 2r times 2πr, or 4πr2.
To figure out the volume of a sphere (how much air or water it would take to fill it up), you multiply the radius by itself and then by itself again, and then by pi, and then by 4, and divide the whole thing by 3. So the formula for the volume of a sphere is 4πr3/3.
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