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This page you will teach you about surface area and volume of figures!


[|Good link for the sections below]

[|Surface Area and Volume of figures] __Surface area__- Is the area of the sides of the figure. Saying how much wrapping paper would you need to wrap the figure. __Volume__- Is the amount of space inside the figure. __Surface Area formula__- S=2lw+2wh+2lh __Volume formula__- V= lwh L-length W-width H-height If the cube has a unknown sides like it is labeled Z. Ex. S=6Z² V=Z³
 * __//The basics and fundamentals of finding the Surface Area and Volume of all shapes//__**

__Example Volume problem__: If you are filling a tub of water and you want to know how much water to put in exactly to find the volume of a 3-D shape. If you measure L=23ft W=12ft H= 34ft you take for the surface area and volume and then you plug the number into the equation with the related variables

(23)(12)(34)=9384 cubic ft

__Example Surface Area problem__: If you are wrapping a box for a friend's birthday party. You don't want to get a lot of wrapping paper. You measure the length, width and height of the box. L=7 in. W=4 in. H=3 in. Find the surface area of the box. S=2(7)(4)+2(4)(3)+2(7)(3) 56+ 24+ 42=122 square inches



__//**How to find the surface area and volume of a prism**// Altitude__- In a prism a segment which has two end points in the planes including to the two bases in the prism and the segment is perpendicular to both planes. __Height__- The length of an altitude in the prism. __The surface area of a right prism-__ S= L+2B or S=hp+2B, L- lateral area B- bases p- perimeter h- height. __Cavalieri's Principle__- Two solids have the same height, the area that crosses the two solid is paralle to the base, and both solids have the same area, so that concludes that the two solids have the same volume. __The volume of a prism__- V=Bh

__Example of surface area and volume__ You want to find a the surface area and volume of a carton of ice cream, (shaped like a right rectangular prism). The length-9 in width- 5 in, height- 4 in. find the surface area and volume. S= L+2B or S= hp+2B __Surface Area__ To find the area of the base the length times the width. (9)(5)= 45 in Since you have two bases you do 2(45)= 90 in. now you need to find the perimeter of the lateral faces You use the length, width, and one of the base length 9+5+14= 28 in. So you take the perimeter and times by the height 4(28)= 112 in. so that is the lateral area now you use the equation 112+90= 202 square inches

__Volume__ you take the area of the base times the height Area of the base- 45 in. The height- 4 in. so, you multiply V= 45(4)= 180 cubic inches .


 * //__[[image:ghtkj.jpg link="http://www.flickr.com/photos/pocar/2278413090/"]]

[|History of how Ancient Egyptians created pyramids]

The Volume and surface area of Pyramids__//** __Pyramid__- Is a polyhedron with a base. __Base__- Is a polygon with three or more lateral faces. __Lateral face__- Triangles that have the same vertex. __Vertex of a pyramid__- Where all the points of a triangle meet. All the points of a triangle share the same vertex. __Base edge__- Each lateral face of a pyramid has one common edge with the base. __Lateral edge__- The intersection of two lateral faces. __Altitude__- Is a segment connected to the base and the vertex, this segment is perpendicular to the plane of the base. __Height__- The length of the altitude. __Regular pyramid__- Is the base is a regular shape and all the lateral faces are conguent to one another and isosceles. The lateral face edge and the altitude are congruent. __Slant height__- The length of the altitude of the lateral face of a regular pryamid. __Surface area of a regular pyramid__- S= L- Lateral area, B- base area, p- the perimeter of the base, and //l//- slant height S=L+B or S=1/2(//l//p)+2B __Volume of a pyramid__- B- base h- height V= 1/3(Bh)

__Example of finding surface area of a pyramid__ You are in vactioning in Egypt and you go visit the pyramid of Giza and you want to find the surface area of the pyramid. You got the the base length, width, height, and the slant height of the pyramid from your tour guide book. The equation to find the surface area is S=L+B L= Lateral area B= base It says the base of the pyramid has Length= 12 ft Height= 22 ft Width=12 ft Slant height- 14 ft you have to find the lateral area and the base area. to find the area you do the slant height times the length of a one of the sides of the base. (1/2)12(14)= 84ft² so the area of one side is 84 ft² to find the lateral area of all four sides 4(84)= 336 ft² to find the area of the base you do length time width 12(12)= 144 ft² so the area of the base is 144 ft² Now you plug these two numbers into the equation L= 336 ft² B= 144 ft² 336+144= 480 square feet __Volume__ There is a famous pyramid in Washington D.C there is a famous pyramid that is full of pudding, you want to know how much pudding is in the pyramid. The length- 6 ft. height- 8 ft. width- 6 ft. and a slant height- 5 ft. the altitude is 4 ft. The altitude is the height of the pyramid. The equation for volume of a pyramid is V=1/3(Bh) to find the area of the base so you do length times width. 6(6)= 36 ft.

(1/3)36*4= 48 cubic feet.




 * __[|History of cylinder seals]

The Volume and Surface area of cylinders__** __Cylinder__- Is a solid that has circular regions and is translated on a parallel plane. __Lateral surface__- Connects the two circles together. __Bases__- The circular image formed and the one translated and parallel to the image. __Altitude__- A segment that connects and is perpendicular to both parallel base planes. __Height__- The length of the altitude. __Axis__- A segment joining the centers of the bases __Right cylinder__- The axis is perpendicular to both bases. __Oblique cylinder__- The axis is one perpendicular to the two bases. __Surface area of a right cylinder__- S=L+2B or S=2πrh+2πr² __Volume of a cylinder__- R= radius V= πr²h

__Example of finding the surface area and volume of a cylinder__: You want to find the surface area and volume of a a pop can. The radius of the base is 12 in. The altitude which is the height is 24 in. The surface area equation is S= 2πrh+ 2πr². The volume equation is V=πr²h __Surface Area__ so you take r= radius and h= height into the equation. 2(π)(12)(24)= 1809.56, the exact answer is 576π now you do two times pie times that radius squared 2(π)(12²) 288(π) = 904.78 so you do

Now you take 1809.56 and 904.78 and add them together. 1809.56+3919.11= 5728.67 in squares

__Volume__ you do the area of the base times the height the height is 24in to find the base area A=πr² the radius is 12 in so you do 12²=144 in you times by pie 144(π)= 452.389 cubic inches 144π is the exact answer for the volume.



[|Example problems of finding the surface area and volume of a cone]__** __Cone__- Is three dimensional figure that has a circular base. __Base__- Is a circular and flat area on the bottom the cone. __Lateral surface__- The area going around the cones base. The lateral surface on a cube is curved. __Vertex__- Where the lateral surface comes together a single point in the cone. This is not in the plane of the base. __Altitude__- Is a segment that goes from the vertex to the plane of the base. The segment is perpendicular to the base plane. __Height__- Is the length of the altitude in the cone. __Right cone__- Is where the altitude intersects at the center of the base. __Oblique cone__- Is where the altitude does not intersect at the center of the base. __Surface area of a right cone__- L= lateral area, B= base, //r// = radius, //l// = slant height S=L+B or Sπrl+πr² Volume of a cone- r- radius, h- height, B- base V=1/3(Bh) or V=1/3(πr²h)
 * __The volume and surface area of cones

__Example for finding surface area and volume of a cone:__ You want to make your own party hats for your birthday party. You make you radius of your base measure 2 in and your altitude is your height which you measured 6 in. Find the surface area of the party hat. The surface area equation is S = πrl+πr². l- height r- radius __Surface area__ First you take the radius times the height times pie π(2)(6)= 12π= 37.6991 in then you do pie times the radius squared π(2²) = (2)(2)= 4= 4π= 12.5664 you take the two final answers of 37.6691 and 12.5664 and add them together 37.6991+12.5664= 50. 2355 square inches

__Volume__ You are filling a cone up with water you want to know the exact amount to put in in centimeters. The altitude(height) is 10 cm. The radius is 4 cm. Find the volume of the cone. The volume formula is V= 1/3(πr²h) h- height you do radius squared times the height first 4²= (4)(4)=16(10)= 160 cm now you take 160 and times it by pie 160π= 502.655 cm you take 502.655 and multiply by (1/3) (1/3)502.655= 167.552 cubic cm



__**The volume and surface area of spheres**__ [|Sphere]- Is a the set of points floating in space, that are all the same distance from a certain point, which is known as the center of the sphere. __Annulus__- Is a region between two circles in a plane, they have the same center point but, different radius. __Volume of a sphere__- V= 4/3(πr³) __Surface area of a sphere__- S=4πr²

__Example of finding surface area and volume of a sphere:__ My mom and I went to the grocery store and I bought a pretty princess ball. When we got home my dad wanted to find the surface area of the ball. We measure the radius to be 7 in. The equation of the surface area of a sphere is S=4πr² __Surface area__ first we do the radius squared 7²=(7)(7)= 49 in then we multiply 49 by pie 49π = 153.938 in. so now we do 153.938(4)= 615.752 square in.

__Volume__ Now we want to find the volume of the ball. The radius was 7 in. The equation for volume is V= (4/3)πr³ so now we do the radius cubed 7³=(7)(7)(7)= 343 in now we multiply 343 times pie 343π = 1077.57 in we take 1077.57 and multiply by (4/3) (4/3)1077.57= 1436.76 cubic in.

__**Amazing Three-dimensional Symmetry**

[|Symmetry]

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