1. 10.4 Volumes of Prisms & Cylinders
Geometry

2. Volume Volume – Is the space that a figure occupies. cm3, in3, m3, ft3
Measured in cubic units (it has 3 dimensions)
cm3, in3, m3, ft3
The formulas for right prisms can also be used for oblique prisms

3. With rectangular prisms it is often easiest to use this formula:
Volume = l ● w ● h
For a cube you can use:
Volume = s³

4. I. Finding the volume of a Prism
Prism – 2 congruent parallel bases, sides are rectangles.
V = Bh
Height of Prism
Area of Base
A = bh (Rectangle)
A = ½bh (Triangle)
A = ½ap (Polygon)
Height (h)
Area of Base (B)

5. Ex. 1: Finding the Volume of a rectangular prism
The box shown is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box? What is the volume of the box?
5 ∙ 3 ∙ 4
= 60 u³

6. Ex.1: Find the Volume of the Prism

7. Ex.2: Find the volume of the following
29² - 20² = 441
441 = 21
½ b ∙ h = ½ (20 ∙ 21)
= ½ (420) = 210
B ∙ h = 210 ∙ 40
= 8400 m³
29m a
40m
20m

8. Ex.3: Yet another prism! Find the volume.8 h
60° 4
10in
Sin 60 = h/8
.866 = h/8
6.9 = h
V = Bh
= ½bh • h
= ½(8in) __ • (10in)
= (27.7in2) • (10in)
= 277in3
8in
6.9

9. B ∙ h = 200 m³ B ∙ 20 = 200 B = 10 B = ½ b ∙ h 10 = ½ 8 ∙ h 10 = 4 ∙ h
Ex. 4 The volume of this figure is 200 m³. Find x.
B ∙ h = 200 m³
B ∙ 20 = 200
B = 10
B = ½ b ∙ h
10 = ½ 8 ∙ h
10 = 4 ∙ h
2.5 = h = x X
20m 8m

10. Finding Volumes of prisms and cylinders.
Bonaventura Cavalieri ( ). To see how it can be applied, consider the solids on the next slide. All three have cross sections with equal areas, B, and all three have equal heights, h. By Cavalieri’s Principle, it follows that each solid has the same volume.

11. Cavalieri’s Principle
If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

12. V = Bh II. Volume of a Cylinder Height of cylinder r h
Volume of right cylinder
Area of base: (Circle)
A = r2

13. Ex.5: Find the area of the following right cylinder.
Area of a Circle
V = Bh
= r2 • h
= (8ft)2 • (9ft)
= 64ft2 • (9ft)
= 576ft3
= ft3
16ft
9ft

14. Ex.6: Find the volume of the following composite figure.
Half of a cylinder:
Vc = Bh
= r2•h
= (6in)2 • (4in)
= 452in3
= 452/2 = 226in3
11in
4in
Volume of Prism:
Vp = Bh
= (11)(12)(4)
= 528in3
12in
VT = Vc + Vp
= 226in in3
= 754in3

15. Find the volume of the solid below.
Rectangular prism: 25 ∙ 30 ∙ 25 = cm³
Cylinder hole: (r = 4 cm)
Cylinder = π 4² ∙ 30 = 480π ≈ cm³
18750 – 1508 ≈ cm³