12.3 surface area of pyramids and cones - rjs solutions · 12.3 surface area of pyramids and cones...

12.3 Surface Area of Pyramids and ConesGoal p Find the surface areas of pyramids and cones.

VOCABULARY

Pyramid

Vertex of a pyramid

Regular pyramid

Slant height

Cone

Vertex of a cone

Right cone

Lateral surface

Pyramids are classified by the shapes of their bases.

Copyright © Holt McDougal. All rights reserved. Lesson 12.3 • Geometry Notetaking Guide 333

12.3

Your Notes

LAH_GE_11_FL_NTG_Ch12_325-355.indd 333LAH_GE_11_FL_NTG_Ch12_325-355.indd 333 12/17/09 1:56:46 AM12/17/09 1:56:46 AM

12.3 Surface Area of Pyramids and ConesGoal p Find the surface areas of pyramids and cones.

VOCABULARY

Pyramid A pyramid is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex.

Vertex of a pyramid The vertex of a pyramid is the common vertex of the lateral faces of a pyramid.

Regular pyramid A regular pyramid has a regular polygon for a base and the segment joining the vertex and the center of the base is perpendicular to the base. The lateral faces of a regular pyramid are congruent isosceles triangles.

Slant height The slant height of a regular pyramid is the height of a lateral face of the regular pyramid.

Cone A cone has a circular base and a vertex that is not in the same plane as the base.

Vertex of a cone The vertex of a cone is the point on the cone that is located at a perpendicular distance from the base, called the height of the cone.

Right cone In a right cone, the segment joining the vertex and the center of the base is perpendicular to the base and the slant height is the distance between the vertex and a point on the base edge.

Lateral surface The lateral surface of a cone consists of all segments that connect the vertex with points on the base edge.

Pyramids are classified by the shapes of their bases.


12.3

Your Notes


Your Notes

Find the area of each lateral face of the regular square pyramid.

h 5 97 cm

b 5 90 cm b 5 45 cm12

slantheight,

SolutionUse the Pythagorean Theorem to find

h 5 97 cm

b 5 45 cm12

slantheight, the slant height l.

l2 5 h2 1 1 1 } 2 b 2 2 Write formula.

l2 5 2 1 2 Substitute for h and 1 } 2 b.

l2 5 Simplify.

l 5 Take the square root of each side.

l < Find the positive square root.

Find the area of each triangular face.

A 5 1 } 2 bl Write formula.

< 1 } 2 ( )( ) Substitute for b and l.

< Simplify.

The area of each lateral face is about square centimeters.

Example 1 Find the area of a lateral face of a pyramid

THEOREM 12.4: SURFACE AREA OF A REGULAR PYRAMID

The surface area S of a regular pyramid is the sum of the base area and the lateral area:

S 5 B 1 ,

B Pwhere B is the area of the base, P is the perimeter of the base, and l is the slant height.

334 Lesson 12.3 • Geometry Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes

Find the area of each lateral face of the regular square pyramid.

h 5 97 cm

b 5 90 cm b 5 45 cm12

slantheight,

SolutionUse the Pythagorean Theorem to find

h 5 97 cm

b 5 45 cm12

slantheight, the slant height l.

l2 5 h2 1 1 1 } 2 b 2 2 Write formula.

l2 5 97 2 1 45 2 Substitute for h and 1 } 2 b.

l2 5 11,434 Simplify.

l 5 6 Ï}

11,434 Take the square root of each side.

l < 106.93 Find the positive square root.

Find the area of each triangular face.

A 5 1 } 2 bl Write formula.

< 1 } 2 ( 90 )( 106.93 ) Substitute for b and l.

< 4811.85 Simplify.

The area of each lateral face is about 4811.85 square centimeters.

Example 1 Find the area of a lateral face of a pyramid

THEOREM 12.4: SURFACE AREA OF A REGULAR PYRAMID

The surface area S of a regular pyramid is the sum of the base area and the lateral area:

S 5 B 1 1 } 2 Pl ,

B Pwhere B is the area of the base, P is the perimeter of the base, and l is the slant height.



Your Notes

Find the surface area and lateral area of the regular hexagonal pyramid.

8 in.

18 in.

4 3 in.

Solution First, find the area of the base using the formula for the area of a regular

8 in.

34 in.

polygon, 1 } 2 aP. The apothem a of the

hexagon is inches and the perimeter P is 6 p 5 inches.

So, the area of the base B is 1 } 2 ( )( ) 5

square inches. Then, find the surface area.

S 5 B 1 1 } 2 Pl Surface area of regular pyramid

5 1 1 } 2 ( )( ) Substitute.

5 1 Simplify.

< Use a calculator.

The surface area of the regular hexagonal pyramid is about square inches.

The lateral area is square inches.

Example 2 Find the surface area of a pyramid

THEOREM 12.5: SURFACE AREA OF A RIGHT CONE

The surface area S of a right cone is the sum of the base area and lateral area:

r

S 5 B 1 1 } 2 Cl 5 ,

where B is area of the base, C is the circumference of the base, r is the radius of the base, and l is the slant height.

Remember, the apothem of a polygon is the distance from the center to any side of the polygon.


Your Notes

Find the surface area and lateral area of the regular hexagonal pyramid.

8 in.

18 in.

4 3 in.

Solution First, find the area of the base using the formula for the area of a regular

8 in.

34 in.

polygon, 1 } 2 aP. The apothem a of the

hexagon is 4 Ï}

3 inches and the perimeter P is 6 p 8 5 48 inches.

So, the area of the base B is 1 } 2 ( 4 Ï}

3 )( 48 ) 5

96 Ï}

3 square inches. Then, find the surface area.

S 5 B 1 1 } 2 Pl Surface area of regular pyramid

5 96 Ï}

3 1 1 } 2 ( 48 )( 18 ) Substitute.

5 96 Ï}

3 1 432 Simplify.

< 598.3 Use a calculator.

The surface area of the regular hexagonal pyramid is about 598.3 square inches.

The lateral area is 432 square inches.

Example 2 Find the surface area of a pyramid

THEOREM 12.5: SURFACE AREA OF A RIGHT CONE

The surface area S of a right cone is the sum of the base area and lateral area:

r

S 5 B 1 1 } 2 Cl 5 πr2 1 πrl ,

where B is area of the base, C is the circumference of the base, r is the radius of the base, and l is the slant height.

Remember, the apothem of a polygon is the distance from the center to any side of the polygon.



Find the surface area of the right cone.

3 m

5 mSolution

The base has a radius of meters, and the cone has a slant height of

meters.

S 5 πr2 1 πrl Surface area of right cone

5 π( )2 1 π( )( ) Substitute.

5 π 1 π Simplify.

5 π Add.

< Use a calculator.

The surface area of the right cone is about square meters.

Example 3 Find the surface area of a right cone

1. Find (a) the area of each lateral face

20 mm

30 mmand (b) the surface area of the regular square pyramid.

2. Find the surface area and lateral

6 ft

10 ftarea of the right cone. Round your answers to two decimal places.

Checkpoint Complete the following exercises.


Homework

Your Notes


Find the surface area of the right cone.

3 m

5 mSolution

The base has a radius of 3 meters, and the cone has a slant height of 5 meters.

S 5 πr2 1 πrl Surface area of right cone

5 π( 3 )2 1 π( 3 )( 5 ) Substitute.

5 9 π 1 15 π Simplify.

5 24 π Add.

< 75.4 Use a calculator.

The surface area of the right cone is about 75.4 square meters.

Example 3 Find the surface area of a right cone

1. Find (a) the area of each lateral face

20 mm

30 mmand (b) the surface area of the regular square pyramid.

a. 300 mm2 b. 1600 mm2

2. Find the surface area and lateral

6 ft

10 ftarea of the right cone. Round your answers to two decimal places.

The surface area is about 301.59 ft2.

The lateral area is about 188.50 ft2.

Checkpoint Complete the following exercises.


Homework

Your Notes


12.3 surface area of pyramids and cones - rjs solutions · 12.3 surface area of pyramids and cones...

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