Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry

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1 Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry

2 Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel planes. The other faces called lateral faces, are parallelograms formed by connecting the corresponding vertices of the bases. The segments connecting these vertices are lateral edges.

3 Finding the surface area of a prism The altitude or height of a prism is the perpendicular distance between its bases. In a right prism, each lateral edge is perpendicular to both bases.

4 Finding the surface area of a prism Find the surface area of a right rectangular prism with a height of 8 inches, a length of 3 inches, and a width of 5 inches.

5

6 Using Theorem 12.2

7 Finding the surface area of a pyramid A pyramid is a three dimensional solid in which the base is a polygon and the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The altitude or height of a pyramid is the perpendicular distance between the base and the vertex.

8 More on pyramids A regular pyramid has a regular polygon for a base and its height meets the base at its center. The slant height of a regular pyramid is the altitude of any lateral face. A nonregular pyramid does not have a slant height.

9 Pyramid Arena

10 Finding the Area of a Lateral Face Architecture. The lateral faces of the Pyramid Arena in Memphis, Tennessee, are covered with steal panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.

11 Surface Area of a Regular Pyramid Formula to find the surface area S of a regular pyramid is: S = B + ½ Pl, where B is the area of the base, P is the perimeter of the base, and l is the slant height. S = 4(LA) + BA, where LA is the area of one lateral side and BA is the area of the base. Formula for LA is (½ base x height)

12

13 Surface Area of a Regular Pyramid Slant Height = ft Base Area = 300 x 300 = 90,000 ft 2 Base Perimeter = 4(300) = 1200 ft Total Surface Area = 90,000 + ½ (1200)(354.42) 302,652 ft 2 S = B + ½ Pl

14 Example Find the total surface area of a regular pyramid with a square base if each edge of the base measures 16 inches, the slant height of a side is 17 inches and the altitude is 15 inches. The perimeter of the base is 4s since it is a square. p = 4(16) = 64 inches The area of the base is s 2. B = 16 2 = 256 inches 2 T. S. A. = ½ (64)(17) = 800 in 2 S = B + ½ Pl

15 Surface Area of Cylinders Surface Area of Cones Geometry

16 Finding the surface area of a cylinder A cylinder is a solid with congruent circular bases that lie in parallel planes. The altitude, or height of a cylinder is the perpendicular distance between its bases. The radius of the base is also called the radius of the cylinder. A cylinder is called a right cylinder if the segment joining the centers of the bases is perpendicular to the bases.

17 Surface area of cylinders The lateral area of a cylinder is the area of its curved surface. The lateral area is equal to the product of the circumference and the height, which is 2 rh. The entire surface area of a cylinder is equal to the sum of the lateral area and the areas of the two bases.

18 Finding the Surface Area of a Cylinder Find the surface area of the right cylinder.

19 Finding the height of a cylinder Find the height of a cylinder which has a radius of 6.5 centimeters and a surface area of square centimeters.

20 Finding the Surface Area of a Cone A circular cone, or cone, has a circular base and a vertex that is NOT in the same plane as the base. The altitude, or height, is the perpendicular distance between the vertex and the base. In a right cone, the height meets the base at its center and the slant height is the distance between the vertex and a point on the base edge.

21 Theorem Surface Area of a Right Cone The surface area S of a right cone is S = r 2 + rl, where r is the radius of the base and l is the slant height

22 Finding the surface area of a cone To find the surface area of the right cone shown, use the formula for the surface area. S = r 2 + rl Write formula S = (4)(6) S = Substitute Simplify S = 40 Simplify The surface area is 40 square inches or about square inches.

23 Surface Area of Spheres Geometry

24 Finding the Surface Area of a Sphere The point is called the center of the sphere. A radius of a sphere is a segment from the center to a point on the sphere. A chord of a sphere is a segment whose endpoints are on the sphere.

25 Finding the Surface Area of a Sphere A diameter is a chord that contains the center. As with all circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.

26 Theorem: Surface Area of a Sphere The surface area of a sphere with radius r is S = 4 r 2.

27 Finding the Surface Area of a Sphere Find the surface area. When the radius doubles, does the surface area double?

28 S = 4 r 2 = = 16 in. 2 S = 4 r 2 = = 64 in. 2 The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16 4 = 64 So, when the radius of a sphere doubles, the surface area DOES NOT double.

29 Finding the Surface Area of a Sphere Baseball. A baseball and its leather covering are shown. The baseball has a radius of about 1.45 inches. a. Estimate the amount of leather used to cover the baseball. b. The surface area of a baseball is sewn from two congruent shapes, each which resembles two joined circles. How does this relate to the formula for the surface area of a sphere?

30 Finding the Surface Area of a Sphere

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