kining

=Link to Chapter 9:= =kiin16=

= = = = = = =7.1 Surface Area and Volume= ==l- length w - width h - height s - side=

Surface Area of a Right Rectangular Prism:
S = 2lw+2wh+2lh

Volume of a Right Rectangular Prism:
V = lwh

Surface Area of a Cube:
S = 6s2

Volume of a Cube:
V = s3

Uses for the Surface Area to Volume Ratio:
The surface area to volume ratio displays how an object can have the same volume, but if the length, height, and weight differ from object to object and still add up to the same volume, the surface area can be different, and vice versa. This website [|here] helps explain further, as well as how you can use it.

[|This picture and table] explains how a figure can have the same volume, but much different surface area.

Example: A storing box company wants to have as much advertising space as possible while still using the same amount of material. One design has a l x w x h of 8in.x 6in.x10in. The second design is 11in.x4in.x12in. Q: Which box design would they want to use? A: The second design. The volume is the same, however, the surface area is larger for the second box.

The surface area to volume ratio has real life applications in subjects like science. For example, [|this website] uses the ratio to display how the measuring of cells is done using the ratio.

=7.2 Surface Area and Volume of Prisms=

Definitions:
__Altitude__- A segment in a prism whose two endpoints end at and are perpendicular to both base planes. __Height__- The length of a prism's altitude.

Surface Area of a Right Prism:
S=L+2B or S=hp+2B

Cavalieri's Principle:
Two solids have equal volumes if their heights are the same and if every plane parallel to the bases form cross sections that have equal areas. [|Picture]. These coins are a perfect example of how two solids can have equal areas, even if one is a perfect cylinder, and the other an oblique. [|This website] has an interactive picture to show how you can move the top point of a triangle without changing the area. Another website [|here] puts area and cavalieri's principle together.

Volume of a Prism:
V=Bh

Example Problems:
1. Find the surface area of a right rectangular prism with the given dimensions l=5 w=7 h=2

2. Find the volume of a right rectangular prism with the given dimensions

B=9 sq cm h=5 cm

=7.3 Surface Area and Volume of Pyramids=



Definitions:
Pyramid- A polyhedron containing three or more lateral faces connected to a base. Base- The polygon opposite the vertex. Lateral Faces- The vertex of the pyramid connects the triangular lateral faces. Vertex of the Pyramid- The vertex that connects the lateral faces. Base Edge- The edge of a lateral base that has three lateral faces in common. Lateral Edge- The intersection of two lateral faces. Altitude- A segment perpendicular to the plane base and the vertex. Height- The length of a pyramid's altitude. Regular Pyramid- Has congruent isosceles triangles as lateral faces and a regular polygon as a base. Slant Height- The length of an edge of a lateral face on a regular pyramid.

Surface Area of a Regular Pyramid: S=L+B or S=1/2lp+B

Volume of a Pyramid: V=1/3Bh

=7.4 Surface Area and Volume of Cylinders= Definitions: Cylinder- A solid with two circular regions parallel to each other connected by a lateral surface. Lateral Surface- What connects the circular regions of a cylinder. Bases- The faces of the circular regions of a cylinder. Altitude- The segment that ends at both bases of a cylinder. Height- The length of a cylinder's altitude. Axis- The segment of a cylinder that joins the center of its two bases. Right Cylinder- A cylinder whose axis is perpendicular to its bases. Oblique Cylinder- A cylinder whose axis is not perpendicular to its bases.

Surface Area of a Right Cylinder:
S=L+2B or S=2(3.14)rh+2(3.14)r

Volume of a Cylinder:
V=Bh or V=(3.14)rh



=7.5 Surface Area and Volume of Cones= Definitions: Cone- A solid whose shape is determined by a segment running perpendicular to its base and meeting at an endpoint. Base- The base of a cone is circular. Lateral Surface- The curved face that connects the base to the vertex. Vertex- The endpoint perpendicular to the base of a cone. Altitude- The perpendicular segment of a cone forming from the vertex to the base. Height- The length of a cone's altitude. Right Cone- A cone is a right cone if its altitude intersects the base at the center. Oblique Cone- A cone is and oblique cone if its altitude does not intersect the base at the center.



Surface Area of a Right Cone:
S=L+B or S=(3.14)rl+(3.14)r

Volume of a Cone
V=1/3Bh or V=1/3(3.14)

=7.6 Surface Area and Volume of Spheres=

Definitions: Sphere- A solid globe-like figure who at all points is at an equal distance from its center. Annulus- A division (or slice) of a sphere.

Volume of a Sphere
V=4/3(3.14)r

Surface Area of Sphere
S=4(3.14)r

Question
The envelope of a hot-air balloon is 54 feet in diameter when inflated. The cost of the fabric used to make the envelope is $1.31 per square foot. Estimate the total cost of the fabric for the balloon envelope.

Solution
First estimate the surface area of the inflated balloon envelope. The balloon is approximately a sphere with a diameter of 54 feet, so the radius is 27 feet. S=4(3.14)r{squared} =4(3.14)(27){squared} =4(3.14)(729) =291(3.14)= approximately 9160.9 square feetNow multiply the surface area of the fabric by the cost per square foot to find the approximate cost of the fabric. 9160.9 square feet X $1.31 per square foot = approximately $12,000

These websites may help you understand the surface area: 1. [|Surface Area of Spheres] 2. [|Surface Area Formulas]