Surface Area of Pyramids & Cones

Section 11-3

Objectives• Find the surface area of pyramids• Find the surface area of cones

All About Pyramids• Pyramid - a polyhedron in which one

face (the base) can be any polygon and the other faces (lateral faces) are triangles that meet at a common vertex

• Named by the shape of the base• Altitude - the perpendicular segment

from the vertex to the plane of the base• Height (h) - the length of the altitude

Pyramids Ctd.• Regular pyramid - base is a regular

polygon & lateral faces are congruent isosceles triangles

• Slant height (l) - length of the altitude of a lateral face

Formulas

Find the surface area of a square pyramid with base edges 7.5 ft and slant height 12 ft.

The perimeter p of the square base is 4 X 7.5 ft, or 30 ft.

You are given = 12 ft and you found that p = 30, so you can find the lateral area.

L.A. = p Use the formula for lateral area of a pyramid.12

= (30)(12) Substitute.12

= 180 Simplify.

Find the area of the square base.

Because the base is a square with side length 7.5 ft,B s2 7.52 56.25.

S.A. = L.A. B Use the formula for surface area of a pyramid.

= 180 56.25 Substitute.

= 236.25 Simplify.

The surface area of the square pyramid is 236.25 ft 2.

(continued)

You try• Find the S.A. of a square pyramid

w/ base edges of 5m and slant height of 3m.

• 55m2

Find the lateral area of the hexagonal pyramid below. Round your answer to the nearest whole number.

Use the formula L.A. = p to find the lateral area of the pyramid.12

The altitude of the pyramid, apothem of the base, and altitude of a lateral face form a right triangle, so you can use the Pythagorean Theorem to find the slant height .

= 202 + (4 3)2 = 400 + 48 = 448

L.A. = p Use the formula for lateral area.12

= (48)( 448) Substitute.12

The lateral area of the hexagonal pyramid is about 508 m2.

507.98425 Use a calculator.

(continued)

All About Cones• A cone is pointed like a pyramid, but its

base is a circle.• Altitude - the perpendicular segment

from the vertex to the center of the base.

• Height - the length of the altitude• Slant height (l) - distance from the

vertex to a point on the edge of the base.

Formulas

= r + r 2 Substitute the formulas for L.A. and B.

Find the surface area of the cone in terms of .

S.A. = L.A. + B Use the formula for surface area of a cone.

= (5)(13) + (5)2 Substitute.

= 65 + 25 Simplify.

= 90

The surface area of the cone is 90 in.2.

Your turn• The radius of the base of a cone is

22m. Its slant height is 10m. Find S.A. in terms of .

• 704 m2

Use the formula L.A. = r to find the lateral area of the cone.

Leandre uses paper cones to cover her plants in the early spring. The diameter of each cone is 1 ft, and its height is 1.5 ft. How much paper is in the cone? Round your answer to the nearest tenth.

The cone’s diameter is 1 ft, so its radius r is 0.5 ft.

The altitude of the cone, radius of the base, and slant height form a right triangle. Use the Pythagorean Theorem to find the slant height .

= 0.52 1.52 = 0.25 2.25 = 2.5

Find the lateral area.

L.A. = r Use the formula for lateral area of a cone.

= (0.5) 2.5 Substitute 0.5 for r and 2.5 for .

The lateral area of the cone is about 2.5 ft.2

2.4836471 Use a calculator.

(continued)

Your turn• Find the L.A. of a cone w/ radius 15

in and ht = 20 in.• 1178 in2