Section 9.4 Volume and Surface Area

Section 9.4Volume and Surface Area

What You Will Learn • Volume • Surface Area

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.

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.

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

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

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

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.

Example 1: Volume and Surface Area • Solution

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.

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.

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

Euler’s Polyhedron Formula • Number of vertices number offaces number of edges = 2 + –

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

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.

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

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

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.

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

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

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

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

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

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.

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

Section 9.4 Volume and Surface Area