A Sphere is a three-dimensional ball-shaped object. In order to be a true sphere, an object must be completely symmetrical, meaning that any individual point on its surface will be exactly the same distance from the center as every other point. Unlike most other three-dimensional geometrical objects every sphere is also perfectly round, meaning that there are no edges, no corners, and no faces.

Fun Facts

• Compared to any other three-dimensional shape a sphere has the highest volume held within the smallest surface area.
• Spheres occur in nature most frequently on a very small scale. Bubbles and water droplets are just two examples of naturally occurring spheres.
• Many people don’t realize it, but the Earth is not technically a sphere! It is a spheroid. This means that all points are nearly, but not quite, equidistant from the center.

Calculating Volume

The equation used to calculate the volume of a sphere is: V = 4/3 π r3.

In this equation the “V” is equal to the sphere’s volume, which is the unknown. The “r” represents its radius, which must already be known in order to determine volume. As most students of mathematics, and most fans of pastry humor, already know, the symbol “π” represents the number Pi, which can be roughly estimated as 3.14.

In order to make sense of this equation, let’s use an example. The sphere in question is a small toy ball with a radius of two inches. The equation would thus read: V = 4/3 π 23. Students using calculators can now simply plug in the equation to get the volume. It should come out to around 33.51.

Without a calculator solving for “V” will require a little bit of extra leg-work. The order of operations tells us that the first step to take is to cube the radius, in this case 2. This simply means it must be multiplied by itself three times (2x2x2) to yield an answer of 8. The equation should now look like:

V = 4/3 π x 8, or V = 4/3 (8) π. Next, multiply the radius cubed by 4/3. 4/3 x 8 = 32/3, or roughly 10.67. For most purposes the equation can be left reading V = 8/3π . If a numerical estimate is necessary, though, simply multiply 8/3, or 10.67, by 3.14 to get the final answer of V = 33.51.

Calculating Surface Area

The surface area of a sphere is its outer boundary. It can be calculated using the equation: surface area = 4π r2. For simplicity’s sake let’s call the surface area “S.” As with the equation to calculate volume reviewed above, “r” is equal to the radius of the circle, which must be known in order to find its surface area.

Given the same example of a ball with a radius of 2 inches and determine it’s surface area the initial equation would read: S = 4π 22. The first step is to square the radius. If r = 2, and 2 x 2 = 4, then r2 = 4. The equation should now be: S = 4π(4). Solving for the exactly value would require multiplying 4 x 4 x π, providing an answer of S = 16π . If a numerical approximation is more appropriate, multiply 16 x 3.14 to get S = 50.24.

Easy, right?