Section 9.4 Volume and Surface Area

1. Section 9.4Volume 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. Figure Formula Diagram Rectangular Solid V = lwh Cube V = s3 Cylinder V = πr2h Cone Sphere Volume Formulas h l w s s s r h h r

6. Figure Formula Diagram Rectangular Solid SA=2lw + 2wh +2lh Cube SA= 6s2 Cylinder SA = 2πrh + 2πr2 Cone Sphere Surface Area Formulas 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 πkey on yourcalculator and roundyour answer to thenearest 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 πkey on yourcalculator and roundyour answer to thenearest 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 πkey on yourcalculator and roundyour answer to thenearest 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 offaces number of edges = 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 Cube: 6 faces, 8 vertices, 12 edges Octahedron: 8 faces, 6 vertices, 12 edges Dodecahedron: 12 faces, 20 vertices, 30 edges Icosahedron: 20 faces, 12 vertices, 30 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.