1. Pyramid – a polyhedron with one base (a polygon) and triangular lateral faces that meet at a common vertex. Regular Pyramid – a pyramid with a regular base, and the segment joining the vertex and center of the base is perpendicular to the base. Slant Height – Height of the lateral face of a regular pyramid.

2. EXAMPLE 1 Find the area of a lateral face of a pyramid SOLUTION Use the Pythagorean Theorem to find the slant height l. l 2 =15 2 +8 2 Write formula. l 2 = h 2 +( b) 2 1 2 A regular square pyramid has a height of 15 centimeters and a base edge length of 16 centimeters. Find the area of each lateral face of the pyramid Substitute for h and b. 1 2

3. EXAMPLE 1 Find the area of a lateral face of a pyramid l = 17 Find the positive square root. l 2 = 289 Simplify. A = bl = (16)(17) = 136 square centimeters. 1 2 1 2 The area of each triangular face is ANSWER

5. EXAMPLE 2 Find the surface area of a pyramid SOLUTION Find the surface area of the regular hexagonal pyramid. First, find the area of the base using the formula for the area of a regular polygon, aP. The apothem a of the hexagon is 5√ 3 feet and the perimeter P is 6 10 = 60 feet. So, the area of the base B is (5√ 3)(60) = 150√ 3 square feet. Then, find the surface area. 1 2 1 2

6. EXAMPLE 2 Find the surface area of a pyramid Formula for surface area of regular pyramid. ≈ 679.81 Substitute known values. Simplify. Use a calculator. = 150√ 3 + (60)(14) 1 2 = 150√ 3 + 420 S = B + Pl 1 2 The surface area of the regular hexagonal pyramid is about 679.81 ft 2. ANSWER

7. GUIDED PRACTICE for Examples 1 and 2 1. Find the area of each lateral face of the regular pentagonal pyramid shown. The area of each lateral face is A = bl = (8) (7.3) =29.2 m 2 1 2 1 2 ANSWER

8. GUIDED PRACTICE for Examples 1 and 2 2. Find the surface area of the regular pentagonal pyramid shown. The surface area of the rectangle pentagon pyramid is 256 m 2 ANSWER

9. Cone – circular base and vertex not in the same plane as the base. Height is perpendicular distance from vertex to base.

12. EXAMPLE 3 Standardized Test Practice SOLUTION To find the slant height l of the right cone, use the Pythagorean Theorem. l 2 = h 2 + r 2 l 2 = 8 2 + 6 2 l = 10 Write formula. Substitute. Find positive square root.

13. EXAMPLE 3 Use the formula for the surface area of a right cone. Standardized Test Practice S = πr 2 + πrl = π(6 2 ) + π(6)(10) = 96π Formula for surface area of a right cone Substitute. Simplify. The correct answer is B. ANSWER

14. EXAMPLE 4 Find the lateral area of a cone TRAFFIC CONE SOLUTION To find the slant height l, use the Pythagorean Theorem. l 2 = 18 2 + (5.7) 2, so l ≈ 18.9 inches. The traffic cone can be approximated by a right cone with radius 5.7 inches and height 18 inches. Find the approximate lateral area of the traffic cone.

15. EXAMPLE 4 Find the lateral area of a cone Find the lateral area. Lateral area = πrl = π(5.7)(18.9) ≈ 338.4 Write formula. Substitute known values. Simplify and use a calculator. The lateral area of the traffic cone is about 338.4 square inches. ANSWER

16. GUIDED PRACTICE for Examples 3 and 4 3. Find the lateral area of the right cone shown. The lateral area of the right cone is 1178 yd 2 ANSWER

17. GUIDED PRACTICE for Examples 3 and 4 The surface area of the right cone is 1885 yd 2 ANSWER 4. Find the surface area of the right cone shown.

18. 814: 3-8, 10-15, 22-24