Unit 11: Surface Area and Volume of Solids

By: Vicky Tan and Michelle Wang

Finding Surface Area and Lateral AreaLateral Area: the sum of areas excluding the bases

● The formula for the lateral area of a cylinder is: 2πrh● The lateral area of a cone is πrℓ (ℓ being the slant of the cone)● However, there is no such thing as a lateral area for spheres as a sphere has no bases

Surface Area: the total area of the surface of a three-dimensional object● The surface area of a cylinder is 2πrh+2πr² (the lateral area added with the area of the 2

circle bases)● The surface area of a cone: πrℓ+πr²

lateral area of a cylinder

bases

lateral area

base

ℓ

Finding Volume Volume: The amount of space a 3-dimensional object occupies. ● To find the volume of a prism or cylinder: (area of base)(height)

● The volume of a pyramid or cone: ⅓(area of base)(height)

● The volume of a sphere: 4/3πr³

Surface Area Example

Surface area of a prism+cylinder answer:

(sum of area of all faces)+2πrh+πr²

However, since one of the bases is on top of one of the prisms faces, we need to subtract the prism faces area from the cylinders base area.(SA of prism)-πr²

Correct Work:

Find the SA of the prism (base*height) {14*12(2)+8*12(2)+14*8(2)}-π(4)²and subtract the area of the circle. 752-16π Next, find the SA of the cylinder and 2πrh+πr²�only add one circle area. 2π(4)(8)+π(4)² 80π

Add up both values. (752-16π)+80π

Final Answer: 752+64π

Volume Example

Volume of cone+Volume of cylinder= answer⅓ x area of base of cone

x height + area of circle base

x height

⅓ x πr2 x 4 + x πr2 x 8

⅓ x 9π x 4 + 9π x 8

12π + 72π

3

3

8

4

By using the pythagorean theorem, we can figure out that the slant height of the cone is 5

Final answer=84π

Connections to Other Units

Connection to Unit 10:In order to find the volume of the trapezoidal prism, you must know how to solve for the area of the trapezoid base.

Formula for trapezoids: Base#1 + Base#2 x height 2Correct Work: Use correct volume formula. (area of base x height)

Find area of trapezoid base.. 7 + 4 x 3 2 =16.5 Plug into formula. 16.5 x 8

Final answer: 132

Connections to Other Units

Connection to Unit 7:To find the surface area, you need to know the slant height measurement of the pyramid. In order to find the slant height, you will need to know the pythagorean theorem. Formula: a²+b²=c²

Correct Work:Find area of base 10x8=80

Find the slant height of both triangles 6²+5²=c² 6²+4²=c² c≈7.8 c≈7.2

Find area of the triangles 2(1/2)8x7.8 2(1/2)10 x 7.2 62.4 72

Add all the areas 80+62.4+72

*Remember: the “c” length must be the hypotenuse of the triangle

Final Answer: 214.4

5

6

4

6? ?

Common MistakesWhen solving for the surface area of a shape that has been cut in half, many students forget to add in the base.

For Example: Here is a sphere that has been cut in half. The correct formula to find the surface area of a sphere is: 4πr² Since we are only finding half of the sphere, the correct formula would be (1/2)4πr². However, it doesn’t just stop here. In order to find the correct surface area, you also need to add the area of the circle base. Correct Work: Use the right formula. SA of Half Sphere= (1/2)4πr²+πr²

Substitute in the value for the radius (1/2)4π(6)²+π(6)²

Solve any necessary exponents, multiplication, etc 72π+36π Add values together Final Answer=108π Don’t

forget to add the base in!

Common MistakesThere are many students who confuse the slant height and the regular height of a shape. Be sure to memorize the correct formulas and be able to differentiate between a shapes volume and surface area

For Example: In order to find the volume of this cone, we need to use the formula:

(⅓)Area of Base*Height. Make sure to not confuse the volume formula with the

very similar surface area formula: πrℓ+πr². The ‘ℓ’ stands for the slant height of the

cone. As for the volume formula, instead of being multiplied by its slant height, it is

multiplied by its regular height.

Correct Work:

Use the correct formula. Volume of Cone: (1/3)πr²*height Use Pythag. Theorem to find the height. (3√17)²-3²=c² 153-9=c²

c²=144 c=12

Plug in the height and radius. (⅓)π(3)²*12

Solve Final Answer: 36π

3

3√17

ℓ

h

Real Life Situations VolumeFind the volume of the wedge of cheese.From looking at the cheese, we cantell that it is cut from a cylinder.Formula: (central angle/360)(volume of cylinder)

Correct Work:Determine the central/360. Simplify. 30/360 1/12

Find the volume of the cylinder. (area of base)(height) π(4)²*3 48π

Multiply together. 1/12*48π FInal Answer: 4π

Surface AreaLet’s say you want to paint part of a houseIn order to find out how much paint you have to buy, you must find the surface area that you want to paint. However, you do not want paint on the windows, door, chimney, or roof. FInd the surface area of the area needed to be painted if the sides of the house are 8 by 10.

Total area - windows and door:8x10 - 2x2+3x780 - 4+21

4

3

30°

2

27

3

Final Answer: 55

Thanks for Listening!

happy now

michelle?

holy crap. yes.