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MA_GR_5_13 CHAPTER 9 LINK

-Mr. White, I was absent when the ten extra points were assigned, so you granted me extra time to work on them.

__7.1 Surface Area and Volume__ The objectives of 7.1 are clear: -Explore ratios of Surface Area and Volume successfully. -Develop the concepts of maximizing Volume and minimizing Surface Area successfully.

Surface Area and Volume Formulas:

Surface Area (SA) and Volume (V) of a right rectangular prism with length (l), width (w), and heigh (h) are **SA= 2lw+2wh+2lh** and **V=lwh**

The SA and V of a cube with sides (s) are **SA=6s^2** and **V=s^3**

Example for both: A homeless man is choosing between a few cardboard boxes to live in. He narrows his decision down to 2 boxes. One box has dimensions of h=4, w=2, l=2.5. The second box has dimensions of h=5, w=1, l=4. Which box has greater surface area and thus requires more wallpaper material for the same volume?

Solution: Surface Area= 2x2.5x2 + 2x2.5x4 + 2x2.5x4= 50 unites squared. Volume= 4x2x2.5= 20 units cubed

__7.2 Surface Area and Volume of Prisms__ The objectives of 7.2 are clear: -Define/use a formula for finding the Surface Area of a right prism successfully. - Define/use a formular for finding the Volume of a right prism successfully. - Use Cavalieri's Principle to develop a formular for the Volume of a right or oblique prism successfully.

Definitions:
 * Altitude**- A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes.
 * Height**- The length of an altitude.

Surface Area of a Right Prism:

Surface Area (SA) of a right prism with lateral area (L), bas area (B), perimeter (p), and height (h) is **SA= L+2B** or **SA= hp+2B**

Volume of a Prism:

Volume (V) of a prism with height (h) and base area (B) is **V= Bh**

Cavalieri's Principle:

If two solids both have equal heights and the cross sections formed by every plane parallel to the bases of both solids have equal areas, then the two solids have equal volumes.

Example for Surface Area: What is the surface area of a right prism with lateral area of 60 and base area of 60?

Solution: SA= 60 x 2x60= 7,200 units squared

Example for Volume: What is the volume of a right prism with a height of 10 and a base of 9?

Solution: V= 10 x 9= 90 units cubed

__7.3 Surface Area and Volume of Pyramids__

The objectives of 7.3 are clear: -Define/use a formula for the surface area of a regular pyramid successfully. -Define/use a formular for the volume of a pyramid successfully.

Definitions:
 * Pyramid**- A polyhedron consisting of a **base** (the polygonal face that is opposite the vertex), and three or more **lateral faces** (face of a prism/pyramid that is not the base).
 * Vertex of the Pyramid**- The lateral faces that share a single vertex.
 * Base Edge-** The edge that each lateral face has in common with the base.
 * Lateral Edge**- The intersection of two lateral faces.
 * Altitude-** A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes.
 * Height-** The length of the altitude.
 * Regular Pyramid-** A pyramid whos base is a regular polygon.
 * Slant Height-** The length of an altitude of a lateral face of a regular pyramid.

Surface Area of a Regular Pyramid:

The Surface Area (SA) of a regular pyramid with lateral area (L), base area (B), perimeter of the base (p), and slant height (l) is **SA= L+B** or **SA= (1/2)lp+B**

Volume of a Pyramid:

The Volume (V) of a pyramid with height (h) and base area (B) is **V= (1/3)Bh**

Example for Surface Area: What is the surface area of a regular pyramid with lateral area of 10 and base area of 10?

Solution: SA= 10 x 10= 100 units squared.

Example for Volume: What is the volume of a regular pyramid with a base of 5 and a height of 10?

Solution: V= (1/3)5 x 10= 22.3333333333continued units cubed.

__7.4 Surface Area and Volume of Cylinders__ The objectives of 7.4 are clear: -Define/use a formula for the Surface Area of a right cylinder successfully. -Define/use a formula for the Volume of a cylinder successfully.

Definitions:
 * Cylinder-** A solid that consists of a circular region and its translated image on a parallel plane with a **lateral surface** (surface connecting the bases) connecting the circles.
 * Bases-** The faces formed by the circular region and its translated image.
 * Altitude-** A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes.
 * Height-** The length of the altitude.
 * Axis-** The segment joining the centers of the two bases.
 * Right Cylinder-** If the axis of a cylinder is perpendicular to the bases, then it is this form.
 * Oblique Cylinder-** If the axis of a cylinder is not perpendicular to the bases, then it is this form.

Surface Area of a Right Cylinder:

The Surface Area (SA) of a right cylinder with lateral area (L), base area (B), radius (r), and height (h) is **S= L+2B** or
 * S= 2 x pi x r x h +** **2 x pi x r^2**

The Volume of a Cylinder:

The Volume (V) of a cylinder with radius (r), height (h), and base area (B) is **V= Bh** or **pi x r^2 x h**

Example for Surface Area: What is the surface area of a right cylinder with lateral area of 20, and base area of 50?

Solution: SA= 20 + 2 x 50= 2,000 units squared.

Example for Volume: What is the volume of a right cylinder with base of 50 and height of 100?

Solution: V= 50 x 100= 5,000 units cubed.

__7.5 Surface Area and Volume of Cones__ The objectives of 7.5 are clear: -Define/use the formula for the Surface Area of a cone successfully. -Define/use the formula for the Volume of a cone successfully.

Definitions:
 * Cone-** A three-dimensional figure that consists of a circular **base** (the face formed by its circular region) and a curved **lateral surface** (a lateral face) that connects the base to a single point not in the plane of the base, called the **vertex** (lateral faces meeting together in the same point).
 * Altitude-** A segment that has endpoints in the planes containing the bases and that is perpendicular to both planes.
 * Height-** The length of the altitude.
 * Right Cone-** If the altitude of a cone intersects the base of the cone at its center, the cone is right.
 * Oblique Cone-** If the altitude of a cone does not intersect the base of the cone at its center, the cone is oblique.

Surface Area of a Right Cone:

The Surface Area (SA) of a right cone with lateral area (L), base of area (B), radius (r), and slant height (l) is **S= L+B** or
 * S= pi x r x l+pi x r^2**

Volume of a Cone:

The Volume (V) of a cone with radius (r), height (h), and base area (B) is **V equals** **(1/3)Bh** or **V equals** **(1/3) x pi x r^2 x h**

Example of Surface Area: Find the surface area of a right cone with lateral area of 25 and base area of 55.

Solution: SA= 25 x 55= 1375 units squared.

Example of Volume: Find the volume of a right cone with base of 10 and height of 20.

Solution: V= (1/3) x 10 x 20= 66.66666667 units cubed.

__7.6 Surface Area and Volume of Spheres__ The objectives of 7.6 are clear: -Define/use the formula for the Surface Area of a sphere successfully. -Define/use the formula for the Volume of a sphere successfully.

Definitions:
 * Sphere-** The set of all points in space that are the same distance, r, from a given point known as the center of the sphere.
 * Annulus-** The ring-shaped figure within the sphere.

Volume of a Sphere:

The Volume (V) of a sphere with radius (r) is **V= (4/3)pi x r^3**

The Surface Area (SA) of a sphere with radius (r) is **SA= 4 x pi x r^2**

Example of Surface Area: Find the surface area of a sphere with the radius of 34.

Solution: SA= 4 x pi x 34^2= 14526.72443 units squared.

Example of Volume: Find the volume of a sphere with the radius of 22.

Solution: V= (4/3)pi x 22^3= 44602.2381 units cubed.

For a more detailed and in-depth look: [|http://www.tiem.utk.edu/~gross/bioed/bealsmodules/area_volume.html] A fun site to memorize vocabulary terms and other such games: http://argyll.epsb.ca/jreed/math9/strand3/3107.htm From a professor's way of teaching: http://www.teacherschoice.com.au/Maths_Library/Area%20and%20SA/area_9.htm A different teaching-style from mine, if you were confused: http://argyll.epsb.ca/jreed/math8/strand3/3206.htm**
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