# How to Find the Center & Radius of a Sphere

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Circles and ​spheres​ are universal in nature, and represent two and three-dimensional versions of the same essential form. A circle is a closed curve on a plane, whereas a sphere is a three-dimensional construct. Each of them consists of a set of points that all lie at the same fixed distance from a central point. This distance is called the ​radius​.

Circles and spheres are both symmetrical, and their properties have limitless vital applications in physics, engineering, art, math and every other human endeavor. If you are presented with a math problem involving a sphere, some fairly routine math is all you need to find the center and radius of the sphere as long as you have certain other information about the sphere in hand.

## The Equation of a Sphere with Center and Radius R

The general equation for the area of a circle is

A = πr^2

where ​r​ (or ​R​) is the radius. The widest distance across a circle or sphere is called the diameter (​D​) and is twice the value of the radius. The distance around a circle, known as the circumference, is given by 2π​r​, (or equivalently, π​D​); the same formula holds for the longest path around a sphere.

On a standard ​x​-, ​y​-, ​z​- coordinate system, the center of any sphere can be conveniently placed at the origin (0, 0, 0). This means that if the radius is ​R​, the points (​R​, 0, 0), (0, ​R​, 0) and (0, 0, ​R​) all lie on the surface of the sphere, as do (−​R​, 0, 0), (0, −​R​, 0) and (0, 0,−​R​).

Spheres, like planes, have surface area, which is curved. The Earth and other planets are examples of spheres that have surfaces that are often functionally treated as two-dimensional because any one reasonably-sized portion of the Earth's surface appears as such on the scale of human-being-sized operations.

The surface area of a sphere is given by

A = 4πr^2

and its volume is given by

V = \frac{4}{3}πr^3

This means that if you have a value for the area or the volume, to find the center and radius of the sphere, you can first calculate ​r​, and then you know exactly how far you have to go in a straight line until reaching the center of the sphere, assuming you are not free to establish (0, 0, 0) as the center for convenience.

## Earth as a Sphere

The Earth is not literally a sphere, as it is flattened at the top and bottom thanks in part to spinning around for billions of years. The line forming ts circumference, around the fattest part in the middle, has a special name, the equator.

Problem:​ Given that the radius of Earth is just shy of 4,000 miles, estimate the circumference, surface area and volume.

C = 2π × 4,000 = \text{ about } 25,000 \text{ miles } \\ \,\\ A = 4π × 4,000^2 = \text{ about } 2 × 10^8 \text{ mi}^2 \, \text{ (200 million square miles)} \\ \,\\ A = \frac{4}{3} × π × 4,000^3 = \text{ about } 2.56 ×10^{10} \text{ mi}^3 \,\text{ (256 billion cubic miles)}

#### Tips

• For reference, although the large countries the United States, China, and Canada all appear to take up a significant fraction of the Earth's surface on a globe, each of these nations has an area of between 3 and 4 million square miles, or less than 2 percent of the Earth's surface in each instance.

## Estimating the Volume of a Sphere

As the above example illustrates, if you want to find the volume of a sphere and you do not have an equation of a sphere calculator device handy, you can estimate this by remembering that π is approximately 3 (actually 3.141...) and that (4/3) π is therefore close to 4. If you can get a good estimate of the cube of the radius, you'll be close enough for "ballpark" purposes on the volume.