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Section 9.2 Pyramids, Area, & Volume

Learn about pyramids, their properties, and how to calculate their lateral surface area, total surface area, and volume. Explore the Pythagorean Theorem as it applies to pyramids.

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Section 9.2 Pyramids, Area, & Volume

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  1. Section 9.2Pyramids, Area, & Volume Section 9.2 Nack/Jones

  2. Pyramid • The solid figure formed by connecting a polygon with a point not in the plane of the polygon is called a pyramid. • The polygonal region is called the base & the point is the vertex. • A regular pyramid is a pyramid whose base is a regular polygon and whose lateral edges are all congruent. • The slant height of a regular pyramid is the altitude from the vertex of the pyramid to the base of any of the congruent lateral faces of the regular pyramid. • The line segment from the vertex perpendicular to the plane of the base is the altitude. Ex. 1 p. 415 Section 9.2 Nack/Jones

  3. Pyramid • In the regular pyramid, the distance l is called the slant height of the lateral surfaces of a regular pyramid. • Theorem 9.2.1: In a regular pyramid, the length a of the apothem of the base, the altitude h, and the slant height l satisfy the Pythagorean Theorem, that is l² = a² + h² in every regular pyramid. l h a Section 9.2 Nack/Jones

  4. LateralSurface Area of a Pyramid • Theorem 9.2.2: The Lateral Area L of a regular pyramid with slant height l and perimeter P of the base is given by: L = ½ pl It is simpler to find the area of one lateral face and multiply by the number of faces. Example 2 p. 415 Section 9.2 Nack/Jones

  5. Total Surface Area • Theorem 9.2.3: The total area (surface area) T of a pyramid with lateral area L and base area B is given by ( the sum of the area of all its faces): T = L + B or T = ½ pl + B Example: To find the total area, Find the slant height. Apply Pythagorean Theorem to one face: l ² + 2² = 6² or l = 42 Find Lateral Area: L = ½ pl= ½42 (16) = 32 2 Find the area of the Base: 6 B = 16 l Total Area = 16 + 32 22 6 4 Section 9.2 Nack/Jones

  6. Volume of a pyramid • Theorem 9.2.4: The volume V of a pyramid having a base area B and an altitude of length h is given by: V =1/3 Bh Example 6 p. 419:h = 12, side = 4 Find the area of the base: B = ½aP. Since it is a 30-60-90 triangle, we know that a = 23 B = ½ 23 (64) = 24 3 V =1/3 Bh = 96 3 units3 Section 9.2 Nack/Jones

  7. Another application of the Pythagorean Theorem • Theorem 9.2.5: In a regular pyramid, the lengths of altitude h, radius r of the base and lateral edge e satisfy the Pythagorean Theorem, that is: e² = h² + r² Example 5 p. 418 Section 9.2 Nack/Jones

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