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. = pUse the formula for lateral area of a pyramid = (30)(12)Substitute = 180Simplify.

Find the area of the square base. Because the base is a square with side length 7.5 ft, B  s 2   S.A. = L.A.  BUse the formula for surface area of a pyramid. = 180  56.25Substitute. = Simplify. The surface area of the square pyramid is ft 2. (continued)

You try Find the S.A. of a square pyramid w/ base edges of 5m and slant height of 3m. 55m 2

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 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. = (4 3) 2 = = 448

L.A. = p Use the formula for lateral area = (48)( 448) Substitute The lateral area of the hexagonal pyramid is about 508 m 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. + BUse the formula for surface area of a cone. = (5)(13) + (5) 2 Substitute. = 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  m 2

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. =  = 0.25  2.25 = 2.5

Find the lateral area. L.A. = rUse the formula for lateral area of a cone. = (0.5) 2.5Substitute 0.5 for r and 2.5 for. The lateral area of the cone is about 2.5 ft Use a calculator. (continued)

Your turn Find the L.A. of a cone w/ radius 15 in and ht = 20 in in 2