1. Section 9.4 Volume and Surface Area

2. What You Will Learn
Volume
Surface Area

3. Volume
Volume is the measure of the capacity of a three-dimensional figure.
It is the amount of material you can put inside a three-dimensional figure.
Surface area is sum of the areas of the surfaces of a three-dimensional figure.
It refers to the total area that is on the outside surface of the figure.

4. Volume
Solid geometry is the study of three­dimensional solid figures, also called space figures.
Volumes of three­dimensional figures are measured in cubic units such as cubic feet or cubic meters.
Surface areas of three­dimensional figures are measured in square units such as square feet or square meters.

5. Volume Formulas Sphere Cone V = πr2h Cylinder V = s3 Cube V = lwh
Rectangular Solid
Diagram
Formula
Figure h l w s s s r h h r

6. Surface Area Formulas Sphere Cone SA = 2πrh + 2πr2 Cylinder SA= 6s2
Cube
SA=2lw + 2wh +2lh
Rectangular Solid
Diagram
Formula
Figure h l w s s s r h r h r

8. Example 1: Volume and Surface Area
Determine the volume and surface area of the following three­dimensional figure.
Solution

9. Example 1: Volume and Surface Area
Determine the volume and surface area of the following three­dimensional figure. When appropriate, use the π key on your calculator and round your answer to the nearest hundredths.

10. Example 1: Volume and Surface Area
Solution

11. Example 1: Volume and Surface Area
Determine the volume and surface area of the following three­dimensional figure. When appropriate, use the π key on your calculator and round your answer to the nearest hundredths.

12. Example 1: Volume and Surface Area
Solution

13. Example 1: Volume and Surface Area
Determine the volume and surface area of the following three-dimensional figure. When appropriate, use the π key on your calculator and round your answer to the nearest hundredths.

14. Example 1: Volume and Surface Area
Solution

15. Polyhedra
A polyhedron is a closed surface formed by the union of polygonal regions.

16. Euler’s Polyhedron Formula
Number of vertices
number of edges
number of faces – +
= 2

17. Platonic Solid
A platonic solid, also known as a regular polyhedron, is a polyhedron whose faces are all regular polygons of the same size and shape.
There are exactly five platonic solids.
Tetrahedron:
4 faces,
4 vertices, 6 edges
Dodecahedron:
12 faces,
20 vertices, 30 edges
Icosahedron:
20 faces,
12 vertices, 30 edges
Cube:
6 faces,
8 vertices, 12 edges
Octahedron:
8 faces,
6 vertices, 12 edges

18. Prism
A prism is a special type of polyhedron whose bases are congruent polygons and whose sides are parallelograms.
These parallelogram regions are called the lateral faces of the prism.
If all the lateral faces are rectangles, the prism is said to be a right prism.

19. Prism The prisms illustrated are all right prisms.
When we use the word prism in this book, we are referring to a right prism.

20. Volume of a Prism
V = Bh,
where B is the area of the base and h is the height.

21. Example 6: Volume of a Hexagonal Prism Fish Tank
Frank Nicolzaao’s fish tank is in the shape of a hexagonal prism. Use the dimensions shown in the figure and the fact that 1 gal = 231 in3 to
a) determine the volume of the fish tank in cubic inches.

22. Example 6: Volume of a Hexagonal Prism Fish Tank
Solution
Area of hexagonal base:
two identical trapezoids
Areabase = 2(96) = 192 in2

23. Example 6: Volume of a Hexagonal Prism Fish Tank
Solution
Volume of fish tank:

24. Example 6: Volume of a Hexagonal Prism Fish Tank
b) determine the volume of the fish tank in gallons (round your answer to the nearest gallon).
Solution

25. Pyramid
A pyramid is a polyhedron with one base, all of whose faces intersect at a common vertex.

26. Volume of a Pyramid
where B is the area of the base and h is the height.

27. Example 8: Volume of a Pyramid
Determine the volume of the pyramid.
Solution
Area of base = s2 = 22
= 4 m2
The volume is 4 m3.

28. Homework
P. 521 # 7 – 28, 30 – 48 (x3)