Notes: Surface Area of Pyramids and Cones I. Pyramids The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. 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. 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.

Ex 1a. 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. Lateral area of a regular pyramid

P = 4(14) = 56 cm

Surface area of a regular pyramid

B = 142 = 196 cm2 Ex 1b: Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft. The base area is ퟗ ퟑ ft2. Step 1 Find the base perimeter. The base perimeter is 3(6) = 18 ft.

Step 2 Find the lateral area.

Step 3 Find the surface area. II. Cones

The vertex of a cone is the point opposite the base. 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. 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. Ex 2a: Find the lateral area and surface area of a right cone with radius 9 cm and slant height 5 cm. L = rℓ Lateral area of a cone = (9)(5) = 45 cm2 Substitute 9 for r and 5 for ℓ.

S = rℓ + r2 Surface area of a cone

= 45 + (9)2 = 126 cm2 Substitute 5 for ℓ and 9 for r. Ex 2b: Find the lateral area and surface area of the cone.

l = 17

L = rℓ Lateral area of a right cone

= (8)(17) Substitute 8 for r and 17 for ℓ. = 136 in2

S = rℓ + r2 Surface area of a cone = 136 + (8)2 Substitute 8 for r = 200 in2 and 17 for ℓ. Ex 2c: Find the lateral area and surface area of the right cone. The slant height is 10cm.

L = rℓ Lateral area of a right cone = (8)(10) Substitute 8 for r and 10 for ℓ. = 80 cm2

S = rℓ + r2 Surface area of a cone = 80 + (8)2 Substitute 8 for r and 10 for ℓ. = 144 cm2