Basic Math and Pre-Algebra

PART 3. The Shape of the World

CHAPTER 16. Surface Area and Volume

Spheres

The last category of solids is the hardest to define and one you’ve probably known almost all your life. A sphere is not a polyhedron because it’s not formed from polygons. It’s not a cylinder because it doesn’t have that lateral area. The sphere is the shape you’d produce if you could grab a circle by the ends of a diameter and spin it around that diameter so fast it blurred into something three dimensional. Put another way, it’s a ball.

Officially, a sphere would be defined as the set of all points in space at a fixed distance from a center point. You can think of it as a 3-D circle, or a ball.

Surface Area

The surface area of a sphere depends on the radius of the sphere. The formula SA = 4πr2 gives you the surface area of a sphere of radius r.

An official soccer ball is a sphere with a circumference of 68 to 70 centimeters. That means its diameter is 68/π to 70/π centimeters, so its radius is 34/π to 35/π centimeters. That’s between 10.8 and 11.1 centimeters. The surface area of the official ball will be at least  square centimeters and not more than  square centimeters.

A sphere is the set of all points in space at a fixed distance from a center point.

Volume

The volume of a sphere, like the surface area, depends on the radius of the sphere. The formula for the volume of a sphere is  Volume is measured in cubic units, so it’s not surprising that the radius is cubed. And given how round it is, the presence of n isn’t a surprise, either. Unfortunately, explaining where the 4/3 came from requires more advanced math than I can include here, so you’ll have to just believe me for that part.

The radius of the earth is about 3,959 miles. To find the volume of your home planet, calculate

82 billion times pi is a very large number. It’s approximately 2.6 x 1011 or about 260 billion cubic miles.

CHECK POINT

31. Find the surface area of a sphere with a radius of 8 inches.

32. Find the volume of a sphere with a radius of 12 cm.

33. Find the surface area of a sphere with a diameter of 4 m.

34. Find the volume of a sphere with a diameter of 6 feet.

35. Find the radius of a sphere with a volume of 4500π cubic centimeters.

The Least You Need to Know

• The surface area of a prism is the sum of the areas of the polygons that form the prism. The volume of a prism is the area of the base times the height.

• The surface area of a pyramid is the area of the base plus the areas of the triangles that surround it. The volume of a pyramid is one-third the area of the base times the height.

• The surface area of a cylinder is the area of the two circular bases plus the area of the rectangle that forms the lateral area, or 2πr2 + 2πrh. The volume of a cylinder is the area of the base times the height, or πr2h.

• The surface area of a cone is the area of the circular base plus the lateral area, or πr2 + πrl, where l is the slant height. The volume of a cone is one-third πr2h.

• The surface area of a sphere is 4πr2. The volume of a sphere is 4/3πr3.