1. Lesson 11.2
Prisms pp

2. Objectives:
1. To state Cavalieri’s principle and use it to prove theorems.
2. To derive a formula for the volume of a prism.
3. To find volumes of prisms using the formula.

3. Postulate 11.5
Cavalieri’s Principle. For any two solids, if all planes parallel to a fixed plane form sections having equal area, then the solids have the same volume.

5. Theorem 11.2
A cross section of a prism is congruent to the base of the prism.

6. Theorem 11.2 refers to cross-sections that are parallel to the bases.
A cross section is the intersection of a three-dimensional figure and a plane that passes through the figure that is perpendicular to the altitude.
Theorem 11.2 refers to cross-sections that are parallel to the bases.

8. Theorem 11.3
The volume of a prism is the product of the height and the area of the base: V = BH.

9. EXAMPLE Find the volume of the following regular prism.8
14 3

10. EXAMPLE Find the volume of the following regular prism.
1. Find the apothem
42 + a2 = 82
a2 = 64 – 16 = 48 a 4 8
a = 4 3

11. EXAMPLE Find the volume of the following regular prism.
2. Find the base area
B = ½ap a 4 8
B = ½(4 3)(48)
B = sq. units

12. EXAMPLE Find the volume of the following regular prism.8
14 3
3. Find the volume
V = BH
V = (96 3)(4 3)
V = 4032 cu. units

13. Square: A = s2 Area Formulas
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

14. Practice: Find the volume.
10 cm
12 cm
6 cm

15. Practice: Find the volume.6″ 4″ 8″ 9 5 ″

16. Definition
The semiperimeter of a triangle is one-half of the perimeter of a triangle:
s = 2 c b a +

17. Heron’s Formula s(s – a)(s – b)(s – c) A =
If ABC has sides of lengths a, b, and c and semiperimeter s, then the area of the triangle is
s(s – a)(s – b)(s – c)
A =

18. Practice: Find the volume. The bases are regular hexagons.
24 mm
6 mm

19. 6 mm

20. 6 mm
6 mm
6 mm

21. Find the apothem.
32 + a2 = 62
9 + a2 = 36
a2 = 27
a ≈ 5.2
6 mm a
3 mm

22. B = 1/2ap
= 1/2a(36)
= 1/2(5.2)(36)
= 93.6
6 mm a
3 mm

23. Practice: Find the volume. The bases are regular hexagons.
24 mm
6 mm
V = BH
= 93.6(24)
= mm3

24. Homework pp

25. Find the volume of each prism. 1.
►A. Exercises
Find the volume of each prism. 1. 3 18

26. Find the volume of each prism. 7.
►A. Exercises
Find the volume of each prism. 7. 10
10.4 41

27. Find the volume of each prism. 9.
►A. Exercises
Find the volume of each prism. 9. 10 12 8

28. ■ Cumulative Review 22. Perimeter of a square
State a formula for each.
22. Perimeter of a square

29. ■ Cumulative Review
State a formula for each.
23. Area of a square

30. ■ Cumulative Review 24. Surface area of a cube
State a formula for each.
24. Surface area of a cube

31. ■ Cumulative Review
State a formula for each.
25. Volume of a cube

32. ■ Cumulative Review 26. Surface area of a circular cylinder
State a formula for each.
26. Surface area of a circular cylinder