1. PYRAMIDS AND CONES: VOLUMES, SURFACE AREAS, AND WEIGHTS
Unit 29
PYRAMIDS AND CONES: VOLUMES, SURFACE AREAS, AND WEIGHTS

2. DEFINITIONS
A pyramid is a polyhedron whose base can be any polygon and whose other faces are triangles that meet at a common point called the vertex
Pyramids are named according to the shape of the bases, such as triangular, quadrilateral, pentagonal, hexagonal, and octagonal
In a regular pyramid the base is a regular polygon, and the lateral edges are all equal in length
A circular cone is a solid figure with a circular base and a surface that tapers from the base to a point called the vertex

3. DEFINITIONS
The axis of a circular cone is a line that connects the vertex to the center of the circular base
In a right circular cone the axis is perpendicular to the base
The volume of a pyramid or a right circular cone equals one third the product of the area of the base and height

4. VOLUME EXAMPLE
Find the volume of the Luxor in Las Vegas (square pyramid) shown below:
V =
350 feet
AB = (646)(646) = ft2
Now,

5. COMPUTING HEIGHTS AND BASES
A tent in the shape of a regular pyramid is designed to contain 6.40 cubic meters of airspace. The base of the tent is a square with each side 2.80 meters long. What is the height of the tent?
Compute the area of the base
The base is a square, its area is 2.8 m  2.8 m = 7.84 m2
Substitute the known values into the volume formula and transpose to find the height
h = 2.45 m Ans

6. LATERAL AREAS
Slant heights are used in determining lateral areas of pyramids and cones
The slant height of a regular pyramid is the length of the altitude of any of the lateral faces
The slant height of a right circular cone is the distance from the vertex to any point on the edge of the circular base
The lateral area of a regular pyramid equals one half the product of the perimeter of the base and the slant height
The lateral area of a right circular cone equals one half the product of the circumference of the base and the slant height

7. LATERAL AREA EXAMPLE
Determine the lateral area of the miniature scale steeple (right circular cone) shown below: (10 in tall and 6 in diam)
First the slant height must be determined using the Pythagorean Theorem
Determine the circumference of the base
CB = d = (6") = inches
Now find the lateral area
LA = ½ CB hs
= ½(18.85")(10.44")
= in2 Ans

8. SURFACE AREA EXAMPLE First find the lateral area
The total surface area of a pyramid or cone includes the base plus the lateral area
Determine the total surface area of the pyramid shown below:
First find the lateral area
Now find the total surface area
SA = LA + AB
= 128 in2 + (4"  4") = 144 in2 Ans

9. FRUSTUMS OF PYRAMIDS AND CONES
When a pyramid or a cone is cut by a plane parallel to the base, the part that remains is called a frustum. A frustum of a pyramid and a frustum of a cone are shown below:

10. VOLUMES OF FRUSTUMS
The volume of the frustum of a pyramid or cone is given as:
where
V = volume
h = height
AB = area of the larger base
Ab = area of the smaller base

11. VOLUMES OF FRUSTUMS
The formula for the volume of a frustum of a right circular cone is given as:
where
V = volume
h = height
R = radius of larger base
r = radius of smaller base

12. VOLUME EXAMPLE
Determine the volume in gallons of the gas tank shown below: all measurements are feet
Find AB and Ab
AB = 4’  6’ = 24 ft2
Ab = 2’  4’ = 8 ft2
Now find the volume by substituting in the known values
= ft2

13. VOLUME EXAMPLE
Determine the volume in gallons of the gas tank shown below: all measurements are feet
= ft2

14. LATERAL AREA
The lateral area of the frustum of a regular pyramid is given as:
where
LA = lateral area
hs = slant height
PB = perimeter of larger base
Pb = perimeter of smaller base

15. LATERAL AREA
The formula for the lateral area of the frustum of a right circular cone is given as:
where
LA = lateral area
hs = slant height
R = radius of larger base
r = radius of smaller base

16. SURFACE AREA
The surface area of the frustum of a pyramid or cone equals the sum of the lateral area, the larger base area, and the smaller base area
where
SA = surface area
LA = lateral area
AB = area of larger base
Ab = area of smaller base

17. SURFACE AREA EXAMPLE
Determine the surface area of the circus/elephant pedestal shown below: all measurements are feet
First find the lateral area
LA = hs(R + r)
= (8)( ) = ft2
Determine AB and Ab
AB = (6)2 = ft2
Ab = (3)2 = ft2
Now find the surface area:
SA = LA + AB + Ab
= = ft2 Ans

18. PRACTICE PROBLEMS
Round all answers in the following story problems to two decimal places whenever necessary:
Determine the volume of a right circular cone drinking cup with a base diameter of 6.2 cm and a height of 13.0 cm.
Determine the lateral area of the cup in problem #1.
Determine the surface area of the cup in problem #1.
Find the volume of a pyramidal “rubics cube” with a height of 7 inches and a square base with 6-inch sides.
Determine the lateral area of the pyramid in problem #4.

19. PRACTICE PROBLEMS (Cont)
Determine the surface area of the pyramid in problem #4.
Find the base diameter of a funnel (right circular cone, forget the hole) that has a volume of 840 in3 and a height of 14.2 inches.
Determine the volume of a drinking glass with a top radius of 3.5in and base radius of 2 in and a height of 9 in.
Determine the lateral area of the cone in problem #8. Make sure you use slant height and not height!
Determine the surface area of the cone in problem #8.

20. PROBLEM ANSWER KEY 130.83 cm3 130.16 cm2 160.35 cm2 84 in3 91.39 in2