1. Surface Areas of Polyhedra and Spheres
Lesson 8.8
Surface Areas of Polyhedra and Spheres pp

2. Objectives:
1. To state and apply the formula for surface area of a sphere.
2. To name the regular polyhedra.
3. To state and apply the formulas for surface area of regular polyhedra.

3. Definition
A sphere is the set of all points in space equidistant from a given point.

5. center

6. radius

7. great circle

8. Two great circles separate the sphere into four sections, or lunes.

9. Theorem 8.18
The surface area of a sphere is 4 times the square of the radius: S = 4r2.

10. EXAMPLE 1 Find the surface area of the sphere.8
s = 4r2
s = 482
s = 256 sq. units ≈ 804 sq. units

11. Practice: Find the surface area.
7 m

12. Definition
A regular polyhedron is a polyhedron with faces bounded by congruent regular polygons and with the same number of faces intersecting at each vertex.

13. There are only five regular polyhedra possible:
regular hexagon (cube),
regular tetrahedron,
regular octahedron,
regular dodecahedron, and
regular icosahedron.

14. tetrahedron
hexahedron
octahedron
dodecahedron
icosahedron

17. Theorem 8.19
The surface area of a regular polyhedron is the product of the number of faces and the area of one face: S = nA.

18. EXAMPLE 2 Blake sells calendars in the shape of a regular dodecahedron
EXAMPLE 2 Blake sells calendars in the shape of a regular dodecahedron. To make sure the calendar for a month would fit on a side, he designed the area of one face to cover 3 square inches. Find the total surface area.
The dodecahedron has 12 sides, so:
S = nA = 12(3) = 36 sq. inches

19. Practice: Find the area of a regular octahedron with one face of area 15 square inches.

20. Practice: Find the total surface area of a regular icosahedron with edge 5.4 3
Aequil. triangle = s2 4 3 25 = 5

21. Practice: Find the total surface area of a regular icosahedron with edge 5.
S = nA ÷ ø ö ç è æ 4 3 25
= 20 5 =
≈ sq. units

22. Homework pp

23. ►A. Exercises
Find the total surface area of each sphere. 1. 4

24. ►A. Exercises
5. The surface area of a sphere is  square yards. What is the length of its radius?

25. 7. dodecahedron 9. octahedron
►A. Exercises
Regular Number Number Number
polyhedron of faces of edges of vertices
7. dodecahedron octahedron

26. 11. An octahedron with one face of area 10 square inches
►A. Exercises
Give the surface area of each regular polyhedron.
11. An octahedron with one face of area 10 square inches

27. ►B. Exercises
17. What is the diameter of a softball that has a surface area of 324 square centimeters?

28. 29. The intersection of all faces of a polyhedron
■ Cumulative Review
Identify each set by name.
29. The intersection of all faces of a polyhedron

29. 30. The intersection of two sides of a triangle
■ Cumulative Review
Identify each set by name.
30. The intersection of two sides of a triangle

30. 31. The intersection of two faces of a tetrahedron
■ Cumulative Review
Identify each set by name.
31. The intersection of two faces of a tetrahedron

31. ■ Cumulative Review
Identify each set by name.
32. The intersection of a right pyramid with a plane containing the altitude of the pyramid

32. ■ Cumulative Review
Identify each set by name.
33. The intersection of the lateral surface of a cylinder with a plane parallel to the bases

33. Formulas Square A = s2 Rectangle A = bh Parallelogram A = bh
Triangle A = ½bh
Trapezoid A = ½h(b1 + b2)
Rhombus A = ½d1d2
Equilateral triangle A = s2
Regular polygon A = ½ap
Circle A = r2 4 3

34. Formulas Surface area of a prism S = L + 2B
Lateral surface area right prism L = pH
Surface area of a cylinder S = L + 2B
Lateral surface area right cylinder L = cH
Surface area of a pyramid S = L + B
Lateral surface area reg. pyramid L = ½pl
Surface area of a cone S = L + B
Lateral surface area cone L = ½cl
Surface area of a sphere S = 4r2
Surface area of a reg. polyhedron S = nA