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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ __**Section 1:Exploring the formulas of Surface Area and Volume** Definitions:__ Surface Area: The suface area of any object is the total area of all the exposed or showing surfaces of the object. Volume: The volume of a solid object is the number of the nonoverlapping unit cubes that will exacty fill the interior or inside of that object.

__Surface Area and Volume Formulas__ For a Rectangular Prism:The surface area(S), and the volume(V), of a right rectangular prism with the lenght(L), width(W)and height(H) are: S=2LW+2WH+2LH and V= LWH

For a Cube: The surface area(S), and volume(V) of a cube with side(s) are: S=6s² and V=s³

Would you like an example? Scroll down for an example.

Say a packaging food company is choosing between two box designs with different dimensions. Which box has the greater suface area and do they have the same volume?

Box 1 has a length of 10 inches, width of 4 inches, and height of 4 inches.Box 2 has a length of 5 inches, width of 8 inches, and height of 5 inches. Use the formula for suface area. Box 1=2(10)(4)+2(4)(4)+2(10)(4)=192 square inches Box 2=2(5)(8)+2(8)(5)+2(5)(5)=210 square inches

So Box 2 has greater suface area.

Now for volume: Box 1=(10)(4)(4)=160 cubic inches Box 2=(5)(8)(5)= 200 cubic inches So Box 2 has a greater volume.

Do you get it now? Try your own problem! ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 * __Section 2:Finding the Surface Area of Prisms__**

Definitions: Altitude:The prism has a segment that has endpoints in the planes of the prism and the planes contains the bases, which are both perpendicular to both planes. Height: The altitude's length

__Formulas/Concepts for a Right Prisms__ Surface Area(S), of a right prism with the lateral area(L), the base area(B), height(h), and perimeter(p) S=L+2B or S=hp+2B

__Formulas/ Concepts for Oblique Prisms__ Calalieri's Principle:The solids(prisms) will have equal volume if they have the same height and their cross-sectional areas are equal.

__Volume__ Volume(V) of any prism with height(h), base(B) is: V=Bh

Would you like an example? Scroll down.

Say you are getting a pet snake for your brother's birthday. You are also getting a cage for the snake from the pet store. You are planning to wrap the cage with wrapping paper but you don't don't have very much money left so you want to spend the least amount on the wrapping paper. The cage mesurements are: height: 4ft, width: 3 ft,and length: 5ft. The snake cage is a right prism. First, find the base. L*W 5*315 ft Now, find the surface area of a a right prism. S=L+2B Cage= 5 +2*15= 35 ft Try using the other formula.

__Volume__ If you are moving and a want to find the perfect volume of a box for that 200 year old priceless vase that your Great Aunt Muriel gave to you, you use the formula for volume. The vase is wrapped up tight and shouldn't be able to move in the box. The base is 40 in. and the height is 5 in. V=40(5)=200 cubic inches

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__**Secton 3:Finding the Surface Area and Volume of Pyramids**__ Definitions: Pyramid:a polyhedron, which has 3 or more lateral faces, and a base Base: polygon Lateral Faces:triangles that share or have a single vertex. This is called the vertex of the pyramid. Base edge: the one edge in common or likeness to one of the lateral faces. Lateral edge: Two lateral faces that intersect Altitude: a perpendicular segment from the plane of the base to the vertex Height: length of the altitude in the pyramid Regular Pyramid: the lateral faces are all congruent isoceles triangles and the base is a regular polygon; its lateral edges are also congruent, plus the altitude also intersects the base of the pyramid at its center Slant height: the altitude's length, which is of a lateral face of a regular pyramid

-Note- Pyramids, like their fellows prisms, are named for their base's shape

__Surface Area Formula for Pyramids__ Suface Area(S), of a regular pyramid with the lateral area(L), base area(B), slant height(l), and perimeter of the base(p) is: S=L+B or S=1/2 lp+B

__Volume Formula for Pyramids__ The volume(V) of any pyramid with height(h), and base area(B): V=1/3Bh

Would you like an example? Scroll down.

Say you are making a paper mache pyramid for your social studies class. The formula you are using is S=L+B. Remember, L=lateral edge and B=base. The measurements for your pyramid are: Length: 3 feet Height: 3 feet Width: 4 feet Slant Height: 6 feet

To find the lateral area you do Length(Slant Height)=answer 3(6)=18 feet but you also need the slant height of all four sides so 4(18)= 72 feet

To find the area of a base, you do the length( width) 3(4)=12 feet

Now solve the rest of the problem. Length= 72 feet Base=12 feet S= 72+12=84 feet square feet

But now, after all that hard work, you decided to make the pyramid a pinta instead. So now you need to know the volume of the pyramid so you can filled it up with candy for your social studies class.The altitude is 4 feet and the other measurements are the same as shown above. The formula for the volume is V=1/3 Bh

To find the area of the base you do length*width 3(4)= 12 feet

Now you take the altitude and times it by the base area. 4(12)=48 feet Now you times the answer by 1/3 (1/3)48= 16 cubic feet

Now, go buy some candy for it!! ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^


 * __Section 4: Finding Surface Area and Volume for Cylinders[[image:cylinder.png width="198" height="266" align="right"]]__**

Definitions: Cylinder: a solid that has circular regions and translates to or onto a paralle plane Lateral surface: It connects the translated image of the two circles together Bases: Circular image that is formed and its translated paralle image Altitude: Segement that connects and the paralles base planes is perpendicular to the segement Height: length of the altitude Axis:Segement that joins the centers of the bases Right cylinder: the axis of a right cylinder is perpendicular to both bases Oblique cylinder: the axis of a oblique cylinder is one perpendicular to both of the bases

__Surface Area Formula for a Right Cylinder__ Surface Area(S) of any right cylinder with a lateral edge(L), base area(B), height(h), and the radius(r): S= L+2B or S=2 π rh+2 π r²

__Volume of Any Cylinder__ Volume(V), of any cylinder with base area(B), radius(r), and height(h): V=Bh or V=π r²h

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ __**Section 5:Finding the Surface Area and Volume for Cones**__

Definitions: Cone: a 3-D figure that has a circular base and also has a curved lateral surface which joins to the vertex Base: the flat part of the cone Lateral Surface: The area that goes around the cone Vertex: the place where the lateral surface comes together to one point in the cone figure Altitude: a perpendicular segment in the cone that will go from the vertex to the plane of the base of the cone Height: lenght of the altitude Right cone: the altitude will intersect the cone's base at the center Oblique Cone: altitude doens't intersect unlike the right cone

__Surface Area Formula for a Cone__ Surface Area(S), of any right cone with the lateral edge(L), base of the area(B), slant height(l), and the radius(r) is: S=L+B or S= π r l + π r²

__Volume Formula for a Cone__ Volume(V), of any cone with radius(r), base area(B), and the height(h) is: V=1/3 Bh or V=1/3π r²h

Would you like some help? Try the links below. [|All about cones] [|More help with cones]

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Definitions: Sphere: set of points which are floating in midair, all the same or exact distance from a certain point(hint, it is the center of the sphere) Annulus: an area or region in which it is between the two circles of the plane
 * __Section 6: Finding the Surface Area and the volume of a Sphere__**

__Surface Area Formula for a Sphere__ Surface Area(S), of a sphere with the raduis(r) is: S=4π r²

__Volume Formula for a Sphere__ Volume(V), of a sphere with the raduis(r) is: V=4/3 π r³

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 * __Section 7: 3-D Symmetry__**

[|Symmetry]



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 * __Extra Help__**