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In silence, look at the following images… please do not discuss your ideas until you are asked to share.

2

3

4

5

6

7

8

9

10

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Now, tell me what the concept is:

Pyramids!

(you guys are so smart…now try another one…)

(Please remember to stay quiet until I ask you to share what you think.)

12

13

14

15

16

17

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Now, tell me what the concept is:

Cones!

(so smart again…I knew you could do it.)

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Mr. OwhToday isToday is: Tuesday, February 13: Tuesday, February 13thth, 2007, 2007

Lesson 12.3 – Surface Area of Pyramids & Lesson 12.3 – Surface Area of Pyramids & ConesCones

•NC 27A, 27B, 28A, 28B NC 27A, 27B, 28A, 28B

AnnouncementsAnnouncements::• Quiz for 12.1 – 12.3 is this Thursday (2/15)• You will need a calculator for this chapter

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Lesson 12.3 – Surface Areas of Pyramids & Lesson 12.3 – Surface Areas of Pyramids & ConesCones• Today’s Objectives:

Find the surface area of a pyramid.Find the surface area of a cone.

• Today’s Key Terms: pyramid, regular pyramid, slant height, lateral face, cone, circular cone, right cone, lateral surface

• Materials Needed Today:notebook, geometer/ruler, calculator

• Today’s Geometry Standard: 8, 9

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Hexagonal Pyramid ABCDEFHexagonal Pyramid ABCDEF

> V is the vertex of the pyramid and ABCDEF is the base.> Altitude is the segment from the vertex to the base (h) > Lateral faces are the six triangular faces (ex. VBC)

> are all lateral edges.

V

EFA

B CD

h

O

s

, , ,B A F EV V V V V, D,VC

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1) The base is a regular polygon.

2) All the lateral edges are congruent.

3) All the lateral faces are ≅ isosceles ’s.

4) The height of a lateral face is the

slant height of the pyramid (denoted s).

5) The altitude is the height.

6) The altitude meets the base at its center.

Regular pyramid properties:

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Lateral Area (LA) of a Regular Pyramid: is the area of just the “sides”

LA = ½Ps

NC 27A: theorem

P = Perimeter of the Bases = slant height

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Surface Area (SA) of a Regular Pyramid: is the total area of the “bottom” + all “sides”

SA = B + LA

NC 27B: theorem

B = area of the Base (formula depends on shape)

P = Perimeter of the Bases = slant height

(½Ps)

(area of all the “sides”)

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Example 1: Find the lateral area & surface area. The base is a regular polygon. Answer in both exact form & to the nearest hundredth.

LA= ½ ps P=8+8+8 P=24LA= ½ (24)(12)LA=144cm² 

SA = B + LA

12cm

8cm

8

sB

2 3

4

28 3

4

64 3

4

SA = 16 3 144 171.71cm²

16 3

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Example 2: Find the lateral area & surface area. Answer in both exact form & to the nearest hundredth.

LA = ½ ps P =15+15+15+15 P =60LA = ½ (60)(18)LA = 540in² 

SA = B + LA B = s² B = (15)² B = 225SA = 225 + 540SA = 765in²

18in

1515in

15

15

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ConeCone

A cone has a vertex and a circular base. The axis is the segment that joins the vertex to

the center of the base. If the axis is to the base, then the cone is a

right cone.

Axis

slant height

h

vertex

r

Altitude or height

vertex

Right Cone Oblique Cone

Axis

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1) The base is circular.

2) The altitude is the height.

3) The slant height (denoted s) is the

distance between the vertex and a point

on the base edge.

4) If the height is to the base, then the cone is a right cone.

5) The altitude meets the base at its center.

Right cone properties:

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Lateral Area (LA) of a Right Cone: is the area of just the “sides”

LA = ½(2r)s or

LA = rs

NC 28A: theorem

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Surface Area (SA) of a Right Cone: is the total area of the “bottom” + all “sides”

SA = r² + LA

NC 28B: theorem

(rs)

r = radius of the Bases = slant height

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Example 3: Find the lateral area and surface area of a right cone whose base radius is 8cm and slant height of 15cm.

Answer in both exact form & to the nearest hundredth.

TIP: Try drawing the figure first

Answers:a)

b)

LA cm 2120

SA cm 2184 . cm 2578 05

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Example 4: Find the lateral area and surface area of the following:

SA LB A

4

6LA sr ss s 2 2 24 6

s 2 24 6 52

LA sr ( ) 4 2 13

2 84 13

16 36

unit. 2140 88

2 13

8 13

6 81 13

HW G1 #10 (Due wed 2/15)HW G1 #10 (Due wed 2/15)Page 738 {3–25 odd, 32–35}Page 738 {3–25 odd, 32–35}TIPTIP: Use Pythagorean Theorem for {15 & 21} to find : Use Pythagorean Theorem for {15 & 21} to find slant height & Write ALL answers in slant height & Write ALL answers in exact formexact form too. too.

COPY FIGURES !!!COPY FIGURES !!!

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Example 5: Find the surface area of a hexagonal pyramid with a lateral edge of 13cm and a base edge of 10cm.

SA LB A

SA 150 3 360

cmSA . 2619 81

LA360B 150 3