Introduction: Surface Area of a Sphere

The surface area of a sphere is one of the harder things in math to compute, this instructable will show you how.

A sphere is an object with no sides, vertexes, nor edges. All it is, is a line that has been rotated in 360 degrees in all directions, creating a perfectly round object. Now, not all spheres in real life are perfectly round, in fact, it is impossible for anything to be perfectly round having a constant radius at all points, but those details are usually minute ones that do not matter unless the measurement you are trying to get goes down to micrometers.

Though knowing the surface area of a sphere may not seem important to you, it can really be a useful thing to know. You can use it to approximate the area of earth that is covered by land or water by multiplying the Earth's surface area by 25 or 75 percent. You can use it to figure out how much of a resin you will need to coat some ball, but you will need to know the volume equation for that. Now, those might not be the most useful uses, but maybe you have to do a geography report on some planet or you are trying to paint a sphere and you are tight on money. Either way it is still something that you might need to know.

Step 1: Some Definitions (sort of Like a Materials List)

Now, you will need to know a few definitions before you start.

  • Line: An object that goes between two points
  • Radius(r): Line from center of sphere to surface of sphere
  • Diameter(d): Line that goes from one side of the sphere to the other that goes through the center (2 radii)
  • Circumference: Line that goes around the great circle of the sphere (Diameter x Pi)
  • Great Circle: A circle that is formed when a plane bisects a sphere through the center.
  • Plane: Two dimensional object that is flat but spreads in all directions.
  • Surface Area: The area of the face of a shape
  • Volume: The amount of space an object takes up
  • Pi: The ratio of Circumference/Diameter that's value is 3.1415926535... (3.14 for short in the equation)

Step 2: The Formula

The formula for the surface area of a sphere is 4 x pi x r2 or pi x d2

2 x pi x r is the equation for the circumference of the circle so the surface area of a circle is just that with the 2r (d) part squared. It also can represent a rectangle that has the dimensions d2 x pi

Step 3: Using the Formula

To solve this equation, first square the radius and then multiply the squared radius by four. Then multiply the 4r2 by 3.14 or whatever amount of decimals on Pi your teacher requires. Then add whatever unit the equation started with and square the unit to make the units correct.

Here is an example: Sphere with radius of 6ft

4 x 3.14 x 62
4 x 3.14 x 36
144 x 3.14
314 + 125.6 + 12.56 = 452.16ft2

You have now calculated the surface area of a sphere.

Step 4: A Few Practice Problems

Here are a few to practice with:

1) Find the surface area of a sphere with radius of 8cm

2) Find the surface area of a sphere with diameter of 24cm

3) Find one eighth the surface area of a sphere with radius of 36cm

4) Find how much paint is required to paint a ball that has a diameter of 16 cm with a 1mm coat of paint (ignore the curve of the object; one cm3= one ml 1cm=10mm

Answers - "Answers 1) 803.84cm2 2) 1809.56cm2 3) 2035.75cm2 4) 321.70ml "

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