1. Unit 11: Surface Area and Volume of Solids
By: Sabrina Ma & Varshini Sai Arcota

2. Key Concepts Surface Area: sum of area of all faces of a solid
Bases: two parallel and congruent faces of a solid
Lateral Faces: non-bases of a solid
Lateral Area: sum of areas of lateral faces of a solid
Volume: how much space a solid takes up
Polyhedra: solid with flat faces that are polygons
Prism: polyhedra with two congruent parallel faces and rectangular sides

3. Formulas Surface Area: Prism: add areas of all sides
Cylinder: 2πrh+2πr²
LA: 2πrh
Cone: πrl+πr²
LA: πrl
Sphere:4πr²
Volume:
(area of base • height)
Prism: l•w•h
Cylinder: πr²•h
Cone: ⅓πr²•h
Pyramid: ⅓B•h
Sphere: (4/3)πr³

4. Example #1 Find the lateral area, surface area and volume: LA: 2πr•h
2π4•9= 72π
SA: 2πr•h + 2πr²
2π4•9 + 2π(4)²
72π + 32π
104π
V: B•h
16π•9=144π
How will finding lateral area help you with surface area? Since the lateral area is part of the entire surface area, you just have to add the area of the bases to the lateral area to find the surface area. 4 9

5. Example #2 Find the surface area and volume: SA: ½(4πr²)+2πrh+πr²
2π(6)²+2π6•8+π(6)²
72π+96π+36π=204π
V: ½(⁴/3πr³)+πr²h
⅔π(6)³+π(6)²•8
144π+288π=432π
half-sphere
cylinder 8 6

6. Example #3 Find the volume: Vol of cone: B•h or πr²•h
x(12+8)=12(x+10)
20x=12x+120
8x=120
x=15
_____
____
Find the volume:
Vol of cone: B•h or πr²•h
Small cone: π(9)²•12
81π•12
972π
Large cone: π(15)²•20
225π•20
4500π
Frustum: 4500π-972π
3528π x
Radius of cones’ bases:
►Pythagorean Theorem
12²+y²=15²
144+y²=225
y²=81
y=9
(12+8)²+z²=(15+10)²
20²+z²=25²
400+z²=625
z²=225
z=15 12 y
Frustum 10 8 z
Volume of frustum= Volume of smaller cone subtracted from volume of bigger cone

7. Common Mistakes
forgetting to add or subtract sides when finding surface area
forgetting to figure out the volume for a part of the shape (ex: the half-sphere on a ice cream cone)
adding x and x√3 or x and xπ
using wrong formulas for different solids
forgetting parts of the formula when doing work

8. Connections to Other Units
Unit 11 connects to Unit 10
Unit 10 is about finding areas of shapes
Solids are made of shapes, so knowledge of Unit 10 will help find surface areas of solids for Unit 11
Unit 10 will also help find area of base to find volume
This unit is also connected to Unit 7
Unit 7 is about finding the area and leg lengths of triangles
Many solids contain triangular faces (prisms and pyramids), therefore knowing how to get the area of a triangle is very important
Often times, the triangle can be a base, so you want to be able to find the surface area and volume of the object

9. Real Life Example SA: 2πr•h + 2πr² 2π7•16 + 2π(7)² 224π + 98π
A company wants to make a can for a drink. Figure out, to the nearest whole number, how much liquid can fill the inside of the can, and how much material is used to make the can.
SA: 2πr•h + 2πr²
2π7•16 + 2π(7)²
224π + 98π
322π or 1012 cm²
V: B•h or πr²•h
π(7)²•16
49π•16
784π or 2463 cm³
14 cm
16 cm