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​).

Other Information About Spheres

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.

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