Section 10.5 Volume and Surface Area Math in Our World

Learning Objectives  Find the volumes of solid figures.  Find the surface areas of solid figures.

Polyhedrons A polyhedron is a three-dimensional figure bounded on all sides by polygons. The simplest polyhedra are rectangular solids and cubes, which are bounded on all sides by rectangles or squares.

Polyhedrons A rectangular solid is shown. We can see that you could build the solid out of 12 cubes that are 1 inch on all sides. That tells us that the volume of the rectangular solid is 12 cubic inches. This volume can be obtained by multiplying the length (3 in.), width (2 in.), and height (2 in), giving us our first volume formula.

Volume Formulas The volume of a rectangular solid is the product of the length, width, and height. For a cube, all three dimensions are equal, so the volume is the length of a side raised to the third power (cubed).

EXAMPLE 1 Finding the Volume of a Rectangular Solid Find the volume of the rectangular solid pictured. The length is 4 feet, the width is 30 inches, and the height is 11 feet. First, we need to rewrite 30 inches in terms of feet: SOLUTION Now we use the volume formula with l = 4, w = 2.5, and h = 11: V = lwh = 4 x 2.5 x 11 = 110 ft 3

EXAMPLE 2 Finding the Volume of a Swimming Pool In order to figure out how much chlorine to use, Judi needs to know the capacity of her pool. The pool is a rectangle 18 feet wide and 36 feet long, and has an average depth of 4 feet. How many gallons does it hold? (There are 7.48 gallons in 1 ft 3.) SOLUTION With an average depth of 4 feet, we can think of the pool as a rectangular solid with l = 36 ft, w = 18 ft, and h = 4 ft. V = lwh = 36 x 18 x 4 = 2,592 ft 3

EXAMPLE 2 Finding the Volume of a Swimming Pool SOLUTION Now we use dimensional analysis to convert to gallons: The capacity of the pool is about 19,388 gallons.

Volume Formulas Notice that the volume of a rectangular solid can be thought of as the area of the base (length times width) multiplied by the height. The same is true with this next solid figure. The volume of a right circular cylinder is given by the formula V =  r 2 h where r is the radius of the circular ends and h is the height.

EXAMPLE 3 Finding the Volume of a Cylinder How many cubic inches does a soup can hold if it is a right circular cylinder with height 4 inches and radius 1.5 inches? SOLUTION Using r = 1.5 in. and h = 4 in., we get V =  r 2 h =  x x 4 = 9  ≈ 28.3 in. 3

Volume Formulas The volume of a pyramid is given by the formula

EXAMPLE 4 Finding Volume of a Pyramid The Great Pyramid at Giza was built by the Egyptians roughly 4,570 years ago. It has a square base measuring meters on a side and is meters high. Find its volume. SOLUTION Since the base is a square that is meters on a side, its area is , or 53, square meters. Using the volume formula with B = 53, and h = 138.8, we get

Volume Formulas The next figure we will discuss is the right circular cone. This is similar to a pyramid with a circular base, as shown. Not surprisingly, the volume formula matches the pyramid formula: one-third times the area of the base times the height. But the base is always a circle with area  r 2, so we get the following: where r is the radius of the circular base and h is the height.

EXAMPLE 5 Finding the Volume of a Cone The cups attached to a water cooler on a golf course are right circular cones with radius 1.5 inches and height 3 inches. How many ounces of water do they hold? (One ounce is about 1.8 inches 3.) SOLUTION Using r = 1.5 and h = 3, we get Now we convert to ounces:

Volume Formulas The geometric name for circular object in three dimensions is a sphere. Just as a circle is the set of all points in a plane that are the same distance from a fixed point, a sphere is the set of all points in space that are the same distance away from a fixed point.

Volume Formulas The volume of a sphere is given by the formula where r is the radius of the sphere.

EXAMPLE 6 Finding Volume of a Sphere The famous ball at Epcot center in Orlando has a diameter of 164 feet. Find its volume. SOLUTION To find the volume, we need the radius. The diameter is twice the radius, so the radius in this case is half of 164 feet, or 82 feet. The volume of the ball is about 2,309,565 cubic feet.

Surface Area The area of the outer surface of a three- dimensional figure is called the surface area. Below are the surface area formulas for solid figures:

EXAMPLE 7 Finding the Surface Area of a Cylinder How many square inches of sheet metal are needed to form the soup can in Example 3, which is a right circular cylinder with height 4 inches and radius 1.5 inches? SOLUTION Using r = 1.5 and h = 4, using the formula