Warm UpFind the missing side length of each right triangle with legs a and b and hypotenuse c.

1. a = 7, b = 24

2. c = 15, a = 9

3. b = 40, c = 41

4. a = 5, b = 5

5. a = 4, c = 8

c = 25

b = 12

a = 9

Surface Areas and Volumes of Pyramids and Cones

Sections 12-5, 12-6, 13-2

Pyramids• The slant height of a regular pyramid is the

distance from the vertex to the midpoint of an edge of the base.

• The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.

Lateral and Surface Areas of Pyramids

• The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid.

• The width of the rectangle is equal to the base perimeter of the pyramid.

• L = ½Pl and SA = ½Pl + B

Example: Finding Lateral Area and Surface Area of Pyramids

Find the lateral area and surface area of a regular square pyramid with base edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if necessary.

Cones• The axis of a cone is the segment with endpoints at

the vertex and the center of the base.• The axis of a right cone is perpendicular to the base.• The axis of an oblique cone is not perpendicular to

the base.

Lateral and Surface Areas of Cones

• The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base.

• The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.

• L = ½Pl and SA = ½Pl + B

Example: Finding Lateral Area and Surface Area of Right Cones

Find the lateral area and surface area of the cone.

Example:

Find the lateral area and surface area of the right cone.

Volume of Pyramids and Cones

Example: Finding Volumes of Pyramids

Find the volume of the square pyramid with base edge length 9 cm and height 14 cm.

Example: Finding Volumes of Cones

Find the volume of a cone.

Example 3

Find the volume of the cone.

Lesson Quiz:

Find the volume of each figure. Round to the nearest tenth, if necessary.

1. a rectangular prism with length 25 cm,

width 17 cm, and height 21 cm

2. a cone with diameter 22 cm and height 30 cm

3. a cone with base circumference 8 m and a

height of 7 m

4. a square pyramid with a base side length of

12 m and a height of 15 m

V = 8925 cm3

V = 3801.3 cm3

V = 117.3 3

V = 720 m3