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yansta25 chapter 9 link


 * __[[image:pyrmid.gif]]Surface Area and Volume__**

volume: the amount of space occupied by an object**
 * surface area of a prism: the sum of all the areas of all faces of a prism

//Surface Area and Volume of a Rectangular Prism:// length: the length of a segment is the measure of the distance from one endpoint to the other width:**
 * height: the length of an altitude of a polygon

S = 2lw + 2wh + 2lh V = lwh

//Surface Area and Volume of a Cube://

S = 6s2 V = s3

//Example: Find the surface area and volume of a rectangular prism, if the length is 5, width is 7, and height is 7.5.//

S = 70 + 105 + 75 S = 250 units 2
 * S = 2 (5)(7) + 2 (7)(7.5) + 2 (5)(7.5)

V = (5)(7)(7.5) V = 262.5 units 3**

//Surface Area of a Right Prism://

S = L + 2B or S hp + 2B V = lwh

//Volume of a Prism://

V = bh

//Cavalieri's Principle:// If two solids 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: The net at the bottom is a right triangular prism. What is the surface area?//

//The area of each base is:// =1/2 (2)(22)= 22 //The perimeter of each base is:// =10 + 22 + 15= 47 //The lateral area is then:// L = (30)(47) L = 1410**
 * B**
 * p**
 * L = hp

//Using the base area, perimeter, and lateral area, it is now possible to figure out the surface area of the right triangular prism.//

S = 1410 + 44 S = 1454 units 2
 * S = 1410 + 2 (22)

Surface Area and Volume of a Regular Pyramid



S** =L+B or S= 1/3Bh V = 1/3Bh Example: the roof of a gazebo is a regular octagonal pyramid with a base edge of 4 feet and a slant height of 6 feet. Find the area of the roof. If roofing material costs $4.50 per square foot, find the cost of covering the roof with this material.

L =1/2lp= 1/2(6)(8x4) =96 square feet x $4.50 per square foot= $

Surface Area of a Right Cylinder [|] S =L+2B or S= 2(3.14)rh + 2(3.14)r2

Example: a penn is a right cylinder with a diameter of 19.05 millimeters and a thickness of 1.55 millimeters. Ignoring the raised design, estimate the surface area of a penny.

the radius of a penny is half of the diameter, or 9.525 millimeters. Use the formula for the surface area of a right cylinder.

S = 2(3.14)(9.525(1.55) + 2(3.14)(9.525)2 equals approx. 663.46 square millimeters

Volume of a Cylinder

V= Bh or V = (3.14)r2hn

surface area of a right cone

S =L+B or S= (3.14)rl + (3.14)r2

volume of a cone

V =1/3Bh or V= 1/3(3.14)r2h

Volume of a sphere V = 4/3(3.14)r3

Example: the envelope of a hot-air balloon has a radius of 27 feet when fully inflated. Approx. how many cubic feet of hot air can it hold?

V = 4/3(3.14)(27)3 V = 4/3(19,683)(3.14) V = 26,244(3.14) cubic feet approx. 82,488 cubic feet

surface area of a sphere

S = 4(3.14)r2 S = (12.56)(27)2 S = (12.56)(729) S = 9156.24 units2

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1. surface area of a triangular prism- if its height is 4 base area 10 base perimeter is 28 so S= hp+2B so S=(4)(28)+2(10)

2. volume of a triangular prism- V=Bh so V=(10)(4) = 40

3. volume of a pyramid- V=1/3Bh so the base is 9 and the height is 3, V=1/3(9)(3) = 9

4. surface area of a pyramid- SA=1/2lp+B so the slant height is 4, width and length are 3 and the Base is 12 so SA=1/2(4)(12)+12 = 30

5.volume of a cylinder- V=πr^2h if the radius is 10 and the height is 5 V=π(100)(5) = 500π

6. surface area of a cylinder- SA=2πrh+2πr^2 if the radius is 10 and the height is 5 SA=2π(10)(5)+2π100 = 300π

7. volume of a cone- V=1/3Bh if the base is 50π and the height is 8 V=1/3(50π)(8) = 133π

8.surface area of a cone- SA=πrl+πr^2 if the radius is 4 and the slant height is 2 SA=π(4)(2) + π(16) = 24π

9.volume of a sphere- V=4/3πr^3 if the diameter is 10 V=4/3π(5)^3 = 523.59π

10. surface area of a sphere- SA=4πr^2 if the radius is 5 SA=4π(25)= 314.15π