1. Volume of Cylinders, Pyramids, Cones and Spheres

2. Volume
The volume of a solid is the number of cubic units contained in its interior.

3. Finding Volumes
Cavalieri’s Principle is named after Bonaventura Cavalieri

4. 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.

5. Cavalieri’s Principle
The six pieces maintain their same volume regardless of how they are moved

6. Volume Formulas
Prism - V=Bh, where B is the area of the base and h is the height.
Cylinder - V=Bh=r2h

7. Volume Formulas
Cone - V=1/3Bh

8. Cones Pringles A third of the calories A fact:
If Pringles came in a cone, which was the same height and diameter as the tall tube, it would contain one third of the calories!!!
Why??
Pringles
A third
of the
calories

9. Volume Formulas
Pyramid - V=1/3 Bh, where B is the area of the base and h is the height.
Sphere - V=4/3r3 h

10. Example A = ½ bh Area of a triangle A = ½ (3)(4) Substitute values
Find the volume of the right prism.
A = ½ bh
Area of a triangle
A = ½ (3)(4)
Substitute values
A = 6 cm2
Multiply values -- base
V = Bh
Volume of a prism formula
V = (6)(2)
Substitute values
V = 12 cm3
Multiply values & solve

11. Example A = r2 Area of a circle A = 82 Substitute values
Find the volume of the right cylinder.
A = r2
Area of a circle
A = 82
Substitute values
A = 64 in.2
Multiply values -- base
V = Bh
Volume of a prism formula
V = 64(6)
Substitute values
V = 384in.3
Multiply values & solve
V = in.3
Simplify

12. Example – Cavalieri’s
Find the volume if h = 10 and r = 7

13. Example V = (⅓)Bh = (⅓)l•w•h = (⅓)15•15•22 = (⅓)4950 = 1650cm3
Find the volume of a square pyramid with base edges of 15cm & a height of 22cm.
Square
V = (⅓)Bh
= (⅓)l•w•h
= (⅓)15•15•22
= (⅓)4950
= 1650cm3
22cm
15cm
15cm

14. Example: Find the volume of the following right cone w/ a diameter of 6in.
Circle
V = ⅓Bh
= (⅓)r2h
= (⅓)(3)2(11)
= (⅓)99
= 33 = 103.7in3
11in
3in

15. Example
Recall:
Ex. 5: If the volume of the cylinder is 441π m3, what is the volume of the cone?
Ex. 6: If the radius of the cone in Ex. 5 is 7 m, what is its height?

16. Ex.4: Volume of a Composite Figure
Volume of Cone first!
Vc = ⅓Bh
= (⅓)r2h
= (⅓)(8)2(10)
= (⅓)(640)
= 213.3 = 670.2cm3
10cm
4cm
Volume of Cylinder NEXT!
Vc = Bh
= r2h
= (8)2(4)
= 256 = 804.2cm3
8cm
VT = Vc + Vc
VT = 670cm cm3
VT = cm3

17. Example V = ⅓Bh V = ⅓(r2)h 110 = (⅓)r2(10) 110 = (⅓)r2(10)
The following cone has a volume of 110. What is its radius.
V = ⅓Bh
V = ⅓(r2)h
110 = (⅓)r2(10)
110 = (⅓)r2(10)
11 = (⅓)r2
33 = r2
r = √(33) = 5.7cm
10cm r

18. Example
Find the volume of a sphere with a radius of 3 ft. V = 36 ft3 or ft3

19. Find the radius of a sphere with a volume of 2304  cm3
Example
Find the radius of a sphere with a volume of 2304  cm3