surface area of pyramids & cones section 11-3. objectives find the surface area of pyramids find...
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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
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