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PETO626 CHAPTER 9 7.1: Surface Area and Volume.

Surface Area and Volume: The Surface Area of an object is the total area of all the exposed surfaces of the object. The volume of a solid object is the number of nonoverlapping unit cubes that will exactly fill the interior of the figure.

Formulas of Volume and Surface Area. The Surface Area, S, and volume, V, of a right rectangular prism with length l, width w, and height h are: S=2lw+2wh+2lh and V=lwh.

The surface area, S, and volume, V, of a cube with side s are: S=6s² and V=s³. Or to make it easier, you can find the surface area of a rectangular prism using simply the 4 lateral faces, being L, and the 2 bases, being b, then the equation turns into a much easier; S=L(4)+b(2).

Examples: A rectangular prism with a length of 2cm, a width of 2cm, and a height of 4cm. Find its surface area and volume. Answer: V=lwh or in this equation it would be V=2x2x4, which equals 16cm³. And the surface area equation is S=l(4)+b(2), in this equation the lateral face would be L=lw, which is L=2x2 which equals 4cm. The base would be b=lh, or b=2x4, which is 8cm. So the equation for surface area turns into; S=4(4)+8(2), 4x4=16cm, 8x2=16cm. 16+16=32cm², the answer. This is a bad picture I know, there's a better picture of the same thing if u click on it. Chapter 7.2: Surface Area and Volume of Prisms.

The surface area of a prism may be broken down into 2 parts: the area of the bases, or base area, and the area of the lateral faces, or lateral area.

Since the bases are congruent, the base area is twice the area of 1 base, or 2B, where B is the area of 1 base.

Surface Area of a Right Prism.

The surface area, S, of a right prism with a lateral area L, base area B, perimeter p, and heigh h is: S=L+2B or S=hp+2B.

Example:Find the surface area of a right pentagonal prism with a perimeter of 4, a height of 3, and a base of 6. Answer: S=hp+2B, substitute in what you know and you have S=(3x4)+(6x2), 3x4=12, 6x2=12, 12+12=24, the answer.

Volume of Right Prisms. The volume of a right rectangular prism with length l, width w, and height h is given by V=lwh. Because the base area, B, of this type of prism is equal to lw, you can also write the formula for the volume as V=Bh

Example: A rectangular prisms base has a length of 9, and a width of 6. The prisms height is 4, find the volume. Answer: Once again, just substitute what you know, V=9x6x4. or 54x4=216, the answer.

Cavalieri's Principle: If 2 solids have equal heights and the cross sections formed by every plane parallel to the bases of both solids have equal areas, then the 2 solids have equal volumes.

Volume of a Prism: The volume, V, of a prism with height h and base area B is: V=Bh.

Chapter 7.3: Surface Area and Volume of Pyramids.

A pyramid is a polyhedron consisting of a base, which is a polygon, and 3 or more lateral faces. The lateral faces are triangles that share a single vertex, called the vertex of the pyramid. Each lateral face has 1 edge in common with the base, called the base edge. The intersection of 2 lateral faces is a lateral edge.

Surface Area of a Regular Pyramid: The surface area, S, of a regular pyramid with lateral area L, base area B, perimeter of the base p, and slant heigh l is: S=L+B or S=½lp+B

I personally think the first 1 is easier.

Example: Find the surface area of a regular square pyramid with a lateral area of 16 and a base area of 4. Answer: Easy, just substitute what you know. S=L+B, or S=16+4, 20.

Volume of a Pyramid: The volume, V, of a pyramid with heigh h and base area B is: V=.3xBh .3=1 third. Now volume of a pyramid is a little tricky, because were used to the usual lwh, but nope its different. because a pyramid is pretty much 1 third of a prism. that's why instead of Bh its .3Bh. Example: Find the area of a pyramid with a Base area of 8 and a height of 6. Answer: I truly doubt its a whole number, but lets give it a shot. V=.3Bh, or V=.3x8x6, which equals 14.4, told ya it wasn't whole. Again, if the picture is a tad blurry click on it for a better picture at Flickr. Chapter 7.4: Surface Area and Volume of Cylinders.

A cylinder is a solid that consists of a circular region and its translated image on a parallel plane, with a lateral surface connecting the circles.

Surface Area of a Right Cylinder: The surface area, S, of a right cylinder with lateral area L, base area B, radius r, pi P, and height h is: S=L+2B or S=2Prh+2Pr². Personally I think the 1st one is easier so lets practice with that one. but i'll explain it first. you see, there is only 1 lateral area because the lateral face is round so it only has 1 face, hence the L, the 2B is for the 2 bases that the cylinder has, add the 2 together and u get the surface area.

Example: Find the surface area of a right cylinder with a lateral area of 6 and a base area of 4. Answer: Fill in what you know. S=6+4(2), 4(2)=8 and 8+6=14, the answer.

Volume of a Cylinder: The volume, V, of a cylinder with radius r, height h, pi P, and base area B is: V=Bh or V=Pr²h. Volume in my opinion is sooooooooooooooo easy to get in my opinion that I probably don't even need to do an example, i'll just explain it, u dont need the 2nd base for the volume because that's what the 1st base is for so you just basically take the base and keep stacking it up along the height until you reach the top, that's the area. That picture in my opinion is awesome because it just shows cylinders in open space, but once again, if blurry, click on it for a better picture at Flickr. Chapter 7.5: Surface Area and Volume of Cones.

A cone is a 3-dimensional 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.

Surface Area of a Right Cone: The surface area, S, of a right cone with lateral area L, base of area B, radius r, pi P, and slant height l is: S=L+B or S=Prl+Pr².

Again, 1st one easiest so we're practicing with it.

Example:Scenario: Right cone, lateral area of 16, base area of 29. Find the surface area. Answer: Easy one, 16+29=45, the answer.

Volume of a Cone: The volume, V, of a cone with radius r, height h, pi P, and base area B is: V=1/3Bh or V=1/3Pr²h.

Explanation: A cone is pretty much 1/3 of a cylinder, with a cylinder the volume is Bh, but since a cone is 1/3 of a cylinder u factor in the 1/3 to the equation. Hence the 1/3Bh. Again, if blurry click on it to see a better one at Flickr. Chapter 7.6: Surface Area and Volume of Spheres.

Definition of a Sphere: It's pretty much a bubble. A 3-dimensional circle.

Volume of a Sphere: The volume, V, of a sphere with radius r, and pi P is: V=4/3Pr³.

Surface Area of a Sphere: The surface area, S, of a sphere with radius r, and pi P is: S=4Pr².

I'm sorry with the little explanation but here goes. The radius of a sphere is halfway across the longest part of the sphere. And once again if this picture is blurry, just click on it. 1. Volume of a Triangular Prism. 2. Surface Area of a Triangular Prism. 3. Volume of a Pyramid. 4. Surface Area of a Pyramid. 5. Volume of a Cylinder. 6. Surface Area of a Cylinder. 7. Volume of a Cone. 8. Surface Area of a Cone. 9. Volume of a Sphere. 10. Surface Area of a Sphere.