Drill1) Find the height of a rectangular prism

with a given length of 6 feet a width of 5 feet and a volume of 330 cubic feet?

2) What is the lateral area of a triangular prism where the three sides of the base are 6, 8, and 10 respectively and the height of the prism is 12 feet?

3) What is the surface area of the same triangular prism if the base is a right triangle and 8 & 6 are the legs of the base?

Objectives

• Find the surface area of a pyramid.

• Find the surface area of a cone.

6.3Surface Area Surface Area

of Pyramids and Conesof Pyramids and Cones

Finding the surface area of a pyramid• A pyramid is a polyhedron in which the base is a

polygon and the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The altitude or height of a pyramid is the perpendicular distance between the base and the vertex.

Vocabulary Pyramid: a pyramid consists of one

base and then triangles for lateral faces.

Altitude: is the length of the segment perpendicular from the vertex to the base.

Slant Height: the slant height of a pyramid is the height of one lateral face.

More on pyramids

• A regular pyramid has a regular polygon for a base and its height meets the base at its center. The slant height of a regular pyramid is the altitude of any lateral face. A nonregular pyramid does not have a slant height.

Pyramid Arena

Lateral Area of a Right Regular Pyramid

The lateral area of a pyramid is the sum of all the areas in the lateral faces.

L = ½ lp* Where “l” is the slant height and “p” is

the perimeter of the base.

Vocabulary

Surface Area: The surface area “S” of a pyramid with lateral area “L” and area of a base “B” is:

S = L + B

Surface Area of a Pyramid

Example: The roof of a gazebo is a square pyramid, if one side of the square base is 12 feet and the slant height is 16 feet. Find the lateral area of the roof.

* If the materials cost $3.50 per sq. ft. how much will it cost to build the roof?

Ex. 1: Finding the Area of a Lateral Face• Architecture. The lateral faces of the

Pyramid Arena in Memphis, Tennessee, are covered with steal panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.

Hexagonal Pyramids

• A regular hexagonal pyramid and its net are shown at the right. Let b represent the length of a base edge, and let l represent the slant height of the pyramid. The area of each lateral face is 1/2bl and the perimeter of the base if P = 6b. So the surface area is as follows:

Hexagonal pyramid

S = (Area of base) + 6(Area of lateral face)

S = B + 6( ½ bl)

S = B + (6b)l

S = B + Pl

Substitute

Rewrite 6( ½ bl) as ½ (6b)l.

Substitute P for 6b

Surface Area of a Regular Pyramid

The surface area S of a regular pyramid is:

S = B + ½ Pl, where B is the area of the base, P is the perimeter of the base, and l is the slant height.

Ex. 2: Finding the surface area of a pyramid

• To find the surface area of the regular pyramid shown, start by finding the area of the base.

• Use the formula for the area of a regular polygon,

½ (apothem)(perimeter). A diagram of the base is shown to the right.

Ex. 2: Finding the surface area of a pyramid

After substituting, the area of the base is ½ (3 )(6• 6), or

54 square meters.

3

3

Surface area

• Now you can find the surface area by using 54 for the area of the base, B.3

Vocabulary Cone: is an object that consists of

a circular base and a curved lateral surface which extends from the base to a single point called the vertex.

Finding the Surface Area of a Cone• A circular cone, or cone,

has a circular base and a vertex that is NOT in the same plane as the base. The altitude, or height, is the perpendicular distance between the vertex and the base. In a right cone, the height meets the base at its center and the slant height is the distance between the vertex and a point on the base edge.

Finding the Surface Area of a Cone• The lateral surface of a

cone consists of all segments that connect the vertex with points on the base edge. When you cut along the slant height and like the cone flat, you get the net shown at the right. In the net, the circular base has an area of r2 and the lateral surface area is the sector of a circle.

More on cones . . . • You can find the area of this sector by

using a proportion, as shown below.

Area of sectorArea of circle =

Arc lengthCircumference Set up proportion

Area of sectorl2 =

2r

2lSubstitute

Area of sector = l2 •2r

2lMultiply each side by l2

Area of sector = rl Simplify

The surface area of a cone is the sum of the base area and the lateral area, rl.

Lateral Area of a Cone

The lateral area of a cone is equal to:

* Where “r” is the radius of the base and “l’ is the slant height of the cone.

rlL

Surface Area of a Cone The surface area of a cone is equal

to:

S = L + B

* Where “r” is the radius of the base and “l’ is the slant height of the cone.

2rrlS

Ex. 3: Finding the surface area of a cone• To find the surface area

of the right cone shown, use the formula for the surface area.

S = r2 + rl Write formula

S = 42 + (4)(6) Substitute

S = 16 + 24 Simplify

S = 40 SimplifyThe surface area is 40 square inches or about 125.7 square inches.