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//7.1 Surface Area and Volume//
[|Surface area formulas for right rectangular prisms] S=2//lw + 2wh + 2lh// For a cube the formula is S=6s²

__Volume formulas for right rectangular prisms__ V=//lwh// For a cube the formula is V=s³

__Ratio of surface area to volume__ The surface area to volume ratio is important in helping to find the maximum and minimum volume. It is helpful in maximizing your volume while still having a low production cost. It is also important to help minimize your volume for storage and other things where you have a restricted amount of space.

[|EXAMPLE PROBLEMS] This website has many different problems to help you understand this chapter better. If you do not understand how to do the problem click on the red go to tutor button. It will take you through the steps on how to do the problem.

//7.2 Surface Area and Volume of Prisms//
__DEFINITION OF **Altitude** of a prism__ A segment that has endpoints in planes that are containing both of the bases and is perpendicular to both planes.

__DEFINITION OF **Height** of a Prism__ The height of a prism is the length of the altitude.

__Surface area formulas for a right prism__ S-surface area L-lateral area B-base area P-perimeter H-height

S=L + 2B or S=HP + 2B

__Volume formulas for a prism__ V-volume B-base H-height

V=BH

[|Cavalieri's Principle] If two solids have a equal height and the cross section formed by all of the parallel planes to the bases have the same area, then their volume is the same also.

[|what exactly is a prism?]

The net of a prism is the prism layed flat. This is a net of a triangular prism.
 * __Key concept from class__**

[|NET OF A SQUARE PYRAMID]

//[|7.3 Surface Area and Volume of Pyramids]//
__Definition of **pyramid**__ A polyhedron in which all but one of the polygonal faces intersect at a single point known as the vertex of the pyramid

__Definition of **base**__ The polygonal face that is opposite the vertex

__Definition of **lateral faces**__ The faces of a prism or pyramid that are not base

__Definition of **vertex of the pyramid**__ The point where all lateral faces of the pyramid intersect

__Definition of **base edge**__ An edge that is part of the bas of a pyramid; each lateral face has one edge in common with the base

__Definition of **regular pyramid**__ A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles

__Definition of **slant height**__ In a regular pyramid, the length of an altitude of a lateral face

__Surface area of a regular pyramid__ S-surface area L-lateral area B-base area p-perimeter of the base //l// - slant height

S= L + B or S= .5//l//p + B

__Volume of a pyramid__ V-volume h-height B-base

V=1/3Bh

__Different types of pyramids__ The pyramids are named by the shapes of their bases. (ex. a pyramid with a base that has 10 sides would be a decagonal pyramid)



//[|7.4 Surface Area and Volume of Cylinders]//
__Definition of **Cylinder**__ A solid that consists of a circular region and its translated image in a parallel plane with a lateral surface connecting the circles

__Definition of **lateral surface**__ The curved surface of a cylinder or cone

__Definition of **axis**__ The segment joining the centers of the two bases

__Definition of **right cylinder**__ A cylinder whose axis perpendicular to the bases

__Definition of **oblique cylinder**__ A cylinder that is not a right cylinder

__Surface area of a right cylinder__ S-surface area L-lateral area B-base area r-radius h-height

S = L + 2B or S = 2(3.14)rh + 2(3.14)r²

__Volume of a cylinder__ V-volume r-radius h-height B-base area



//[|7.5 Surface Area and Volume of Cones]//
__Definition of **cone**__ A 3-D figure that consists of a circular base and a curved lateral surface that connects the base to a single point not in the plane of the base, called the vertex

__Definition of **vertex**__ A point where the edges of a figure intersect

__Definition of **right cone**__ A cone in which the altitude intersects the base at its center point

__Definition of **oblique cone**__ A cone that is not a right cone

__Surface area of a right cone__ S-surface area L-lateral area B-base area r-radius //l//-slant height

S = L + B or S = (3.14)r//l// + (3.14)r²

__Volume of a cone__ V-volume r-radius h-height B-base area

V = 1/3 Bh or V = 1/3 (3.14)r²h



//[|7.6 Surface Area and Volume of Spheres]//
__Definition of **sphere**__ The set of points in space that are equidistant from a given point known as the center of the sphere

__Definition of **annulus**__ The region between two circles in a plane that have the same center but different radii

__Volume of a sphere__ V-volume r-radius

V = 4/3(3.14)r³

__Surface area of a sphere__ S-surface area r-radius

S = 4(3.14)r²



Find the volume and surface area of a triangular prism with height of 13 inches, a base length of 6 inches, and a base height of 3 inches.

//__SURFACE AREA__//
__LATERAL AREA__ 1 side- 13• 6 = 18 in² 3 sides- 18• 3 = 54 in²

__BASE AREA__ .5(6•30) = 9 in²

S = L + 2B = 234 + 2 (9) = 252 in²

//__VOLUME__//
V=Bh =(9) (13) = 117 in³

Find the volume and surface area of a square pyramid with base sides of 6 inches and a height of 9 inches.

//__SURFACE AREA__//
__LATERAL AREA__ 1 side- .5 (6•9) = 27 in² 4 sides- 27•4 = 108 in²

__BASE AREA__ 6•6 = 36 in²

S = L + B = 108 + 36 = 144 in²

//__VOLUME__//
V = 1/3 Bh = 1/3 (36•9) = 324 in³

Find the volume and surface area of a cylinder with a diameter of 13 feet and a height of 20 feet.

SURFACE AREA
S = 2(3.14)rh + 2(3.14)r² = 2(3.14)(6.5)(20) + 2(6.5)² = 900.9 in²

VOLUME
V = (3.14)r²h = (3.14)(6.5)²(20) =2653.3 ft³

Find the volume and surface area of a cone with radius of 6 inches, a slant height of 7 inches, and a height of 14 inches.

SURFACE AREA
S = (3.14)r//l// + (3.14)r² = (3.14)(6)(7) + (6)² = 167.88 in²

VOLUME
V = 1/3(3.14)r²h = 1/3(3.14)(6)²(14) =527.52 in³

Find the volume and surface area of a sphere with a radius of 17 feet and a height of 23 feet.

SURFACE AREA
S = 4(3.14)r² = 4(3.14)(17)² = 3629.84 in²

VOLUME
V = 4/3(3.14)r³ = 4/3(3.14)(17)³ =20569.09333 ft³