Exercise

1. Exercise Find the area of a circle with a radius of 4.5 units to the nearest tenth of a square unit 63.6 units2

2. Exercise Find the area of a square with sides of 9.4 units to the nearest tenth of a square unit. 88.4 units2

3. Exercise How many surfaces would you have to find the area of in order to calculate the surface area for a hexagonal prism? 8

4. Exercise How many surfaces would you have to find the area of in order to calculate the surface area for a cylinder? 3

5. Exercise How many surfaces would you have to find the area of in order to calculate the surface area for a sphere? 1

7. Pyramid A pyramid is a three-dimensional figure with a single polygonal base and triangular lateral faces that meet at a common point, known as the vertex of the pyramid.

8. Altitude of a Pyramid The altitude of a pyramid (H) is the perpendicular distance from the vertex to the base. H

9. Slant Height The slant height (l) of a pyramid is the altitude of each triangular face. l

10. Lateral Surface Area The lateral surface area (L) of a pyramid is the sum of the triangular faces.

11. Regular Pyramid A regular pyramid has a regular polygon as its base, and its vertex is directly above the center of the base. Pyramids with squares or equilateral triangles for bases are examples of regular pyramids.

14. L= lateral surface area S = total surface area B = area of the base H = altitude (prism, cylinder, pyramid) l = slant height (pyramid, cone)

19. 12 Formula: Lateral Surface Area of a Regular Pyramid L = pl The lateral surface area of a regular pyramid (L) is equal to half the product of the perimeter of the base (p) and the slant height of the lateral faces (l).

20. Formula: Surface Area of a Regular Pyramid S = L + B The surface area of a regular pyramid (S) is equal to the sum of the lateral surface area (L) and the area of the base (B).

21. L = pl 12 Example 1 Find the lateral surface area and surface area of the square pyramid. 13 m l = 13 m 12 m p = 4(10) = 40 m 10 m 10 m

22. L = (40)(13) 12 = 260 m2 B = s2 = 102 = 100 m2 S = L + B 13 m = 260 + 100 12 m = 360 m2 10 m 10 m

23. l2 = 676 Example 2 Find the slant height of the square pyramid. a = 10 and H = 24 l2 = a2 + H2 = 102 + 242 24 = 100 + 576 = 676 20 = 26 units 10

24. Example What is the surface area of a pyramid with a square base with s = 6 units and l = 7 units? 120 units2

25. Example What is the slant height of a pyramid with a square base with s = 8 units and an altitude of 3 units? 5 units

26. Circular Cone A circular cone is similar to a pyramid but has a circular base.

27. Lateral Surface The lateral surface is the curved surface of the cone. It is not the circular base.

30. 12 Formula: Lateral Surface Area of a Circular Cone L = cl The lateral surface = prlarea of a circular cone (L) is equal to half the product of the circumference of the base (c) and the slant height (l). Substitute for c.

31. Formula: Surface Area of a Circular Cone S = L + B The surface area = prl+ pr2 of a circular cone (S) is equal to the sum of the lateral surface area (L) and the area of the circular base (B). Substitute for L and B.

32. Example 3 Find the surface area of the cone. L = prl S = L + B = p(3)(10) = 30p + 9p = 30p = 39p B = pr2 = 39(3.14) 10 = p(32) = 122.5 3 = 9p

33. Example What is the slant height of a circular cone with d = 24 units and an altitude of 5 units? 13 units

34. Example What is the surface area of a circular cone with d = 24 units and an altitude of 5 units? 300p units2≈ 942 units2

35. Sphere A sphere is a three-dimensional closed surface, every point of which is equidistant from a given point called the center.

36. Chord A chord is a line segment with both endpoints on the sphere.

37. Diameter The diameter is a chord that passes through the center of a sphere to an opposite point on the sphere.

38. Radius A radius is a line segment running from the center of a sphere to a point on the sphere.

39. Plane A plane intersects a sphere on a single point or as a circle.

40. Great Circle A great circle is the largest circle formed by the intersection of a plane and a sphere.

41. Formula: Surface Area of a Sphere S = 4A The surface area of a = 4pr2sphere (S) is equal to four times the area of a great circle (A). Substitute for A.

42. 8 ft. Example 4 Find the surface area of the sphere. S = 4pr2 = 4p(82) = 4p(64) = 256p = 256(3.14) ≈803.8 ft.2

43. Example What is the radius of a circular cone with a surface area of 75p units2 whose slant height is equal to its diameter? 5 units

44. Example Determine the length of each side of the base of a square pyramid with a surface area of 343 m2 if the slant height is three times the length of a side of the base. Each side is 7 m.