Unit 3 Study Guide
Pythagorean Theorem
1.) Solving for side ‘c’ (hypotenuse – the longest side)
You are missing the diagonal of the rectangle which creates a right triangle.
a2 + b2 = c2
52 + 122 = c2
25 + 144 = c2
169 = c2
√169 = c
13 = c
2.) Solving for side ‘a’ or ‘b’ (legs)
You are missing one of the legs of a right triangle.
a2 + b2 = c2
a2 + 182 = 302
a2 + 324 = 900
-324 -324
a2 = 576
a = √576
a = 24
30 ft
18 ft
3.) Area of a square and Pythagorean Theorem
The diagonal of a line is 20 feet. What is the approximate length of each side
of the square?
20
Since the diagonal side is ‘c’, than c2 = 400. You need to find ‘a’ and ‘b’. What
two same numbers squared would be closest to 400?
142 + 142 ≈ 400
196 + 196 ≈ 400
392 ≈ 400
4.) Solving for a missing distance on a coordinate plane
Draw the length and width between the two points on the graph. These
represent the two legs of the right triangle. Solve for side length ‘c’ by using
a2 + b2 = c2
22 + 52 = c2
4 + 25 = c2
29 = c2
√29 ≈ c
5.4 ≈ c
5.) Pythagorean Triples
a2 + b2 = c2
32
42
52
+ =
9 + 16 = 25
A right triangle whose sides are all whole numbers.
Examples.
3, 4, 5
5, 12, 13
6, 8, 10
7, 24, 25
6.) Pythagorean Theorem in 3d Figures
Find the length of EO if EY = 20, YB = 24, and BO = 10.
You need to find the hypotenuse (YO) of the bottom right triangle in
order to use it as a leg in the 3D right triangle.
a 2 + b2 = c 2
a2 + b2 = c2
102 + 242 = c2
202 + 262 = c2
100 + 576 = c2
400 + 676 = c2
Bottom
3D triangle
676 = c2
1076 = c2
Triangle
√676 = c
√1076 = c Length of EO
26 = c
33 ≈ c
Volume of 3D Figures
Cylinders, Cones, and Spheres
Use 3.14 for п to solve for the volume of the following shapes.
1.) Cylinder (stack of circles)
V = ∏r2h
V = ∏ (32)(12)
V = ∏ (9) (12)
V = ∏ (108)
V = 3.14 (108)
V = 339.12
Plug in radius and height
Square the radius first
Multiply radius and height together
Multiply by pi (3.14)
2.) Cone (1/3 the size of a cylinder)
V = ∏r2h
3
V = ∏(102) (30)
3
V = ∏ (100) (30)
3
V = ∏ (3000)
3
V = ∏ (1000)
Plug in radius and height
Square the radius first
Multiply radius and height together
Divide by 3
V = 3,140
3.) Sphere
V = 4∏r3
3
V = 4∏ (9)3
3
V = 4∏ (729)
3
V = 2916 ∏
3
V = 972∏
V = 3,052.08
Plug in radius
Cube the radius
Multiply radius by 4
2916 is divisible by 3 so divide by 3
Multiply by pi (3.14)
4.) Volume of multiple shapes ~ Find total volume
Cone
Hemisphere (half sphere)
3
V = ∏(32) (6)
3
V = ∏(9)(6)
3
V = 54∏
3
V = 18∏
V = 4∏r3 ÷ 2
3
V = 4∏(3)3 ÷ 2
3
V = 4∏(27) ÷ 2
3
V = 108∏ ÷ 2
3
V = 36∏ ÷ 2
V = 56.52 in3
V = 113.04 in3 ÷ 2 = 56.52
V=
∏r2h
Cone + Hemisphere = 56.52 + 56.52 = 113.04 in3