Pyramids And Cones 1qs5t

SOLID MENSURATION PYRAMIDS AND CONES: VOLUMES, SURFACE AREAS, AND WEIGHTS

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 2

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 V

1 AB h where V  volume 3 AB  area of the base h  height 3

VOLUME EXAMPLE 

Find the volume of the Luxor in Las Vegas (square pyramid) shown below:

1 AB h • V = 3

• AB = (646)(646) = 417316 ft2 350 feet

base is 646ft per side

• Now,

3 1 V  ( 417316)(350)  48,686,866.7 ft Ans 4 3

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

1 V  AB h 3 1 3 6.4 m  (7.84 m 2 )h 3 h = 2.45 m Ans 5

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

6

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

hs  32  102  10.44 inches – Determine the circumference of the base CB = d = (6") = 18.85 inches – Now find the lateral area LA = ½ CB hs = ½(18.85")(10.44") = 98.397 in2 Ans 7

SURFACE AREA EXAMPLE 



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

1 LA  PB hs 2

1 (4"  4 sides)(16" )  128 in 2 2 – Now find the total surface area 

SA = LA + AB = 128 in2 + (4"  4") = 144 in2 Ans 8

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:

9

VOLUMES OF FRUSTUMS 

The volume of the frustum of a pyramid or cone is given as: 1 V  h( AB  Ab  AB Ab ) 3 

where    

V = volume h = height AB = area of the larger base Ab = area of the smaller base

10

VOLUMES OF FRUSTUMS 

The formula for the volume of a frustum of a right circular cone is given as:



where    

1 V  πh( R 2  r 2  Rr) 3

V = volume h = height R = radius of larger base r = radius of smaller base

11

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

1 h( AB  Ab  AB Ab ) 3 1  (8) 24  8  (24)(8) 3 = 122.28 ft2

V



 12

VOLUME EXAMPLE 

Determine the volume in gallons of the gas tank shown below: all measurements are feet

= 122.28 ft2

122.28ft 1

3

x

7.5gal 1ft

3

 917.1 gallons

13

LATERAL AREA 

The lateral area of the frustum of a regular pyramid is given as: 1 LA  hs ( PB  Pb ) 2 

where    

LA = lateral area hs = slant height PB = perimeter of larger base Pb = perimeter of smaller base

14

LATERAL AREA 

The formula for the lateral area of the frustum of a right circular cone is given as:

LA  πhs ( R  r ) 

where    

LA = lateral area hs = slant height R = radius of larger base r = radius of smaller base 15

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

SA  LA  AB  Ab 

where    

SA = surface area LA = lateral area AB = area of larger base Ab = area of smaller base 16

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)(6 + 3 ) = 226.19 ft2

– Determine AB and Ab AB = (6)2 = 113.10 ft2

Ab = (3)2 = 28.27 ft2

– Now find the surface area: SA = LA + AB + Ab 17 = 226.19 + 113.10 + 28.27 = 367.56 ft2 Ans

PRACTICE PROBLEMS 

Round all answers in the following story problems to two decimal places whenever necessary:

1.

2. 3. 4.

5.

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.

18

PRACTICE PROBLEMS (Cont) Determine the surface area of the pyramid in problem #4. 7. 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. 8. 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. 9. Determine the lateral area of the cone in problem #8. Make sure you use slant height and not height! 10.Determine the surface area of the cone in problem #8. 6.

19

PROBLEM ANSWER KEY 130.83 cm3 2. 130.16 cm2 3. 160.35 cm2 4. 84 in3 5. 91.39 in2 6. 127.39 in2 7. 15.03 in 8. 219.13 in3 9. 156.03 in2 10. 207.08 in2 1.

20