1. Geometry
A general review

2. Pythagorean Theorem a2 + b2 = c2 Examples: 3, 4 = 5 5, 12 = 13
8, 15 = 17 c a b

3. Three Dimensions
X, Y, Z axis
(x, y, z) z y x

4. Length of a line in 3D
x2 + y2 + z2 = length2
length z y x

5. 45-45-90 Triangle (Pythagorean)
c = a√2 45
√2 a a 90 45 a

6. 30-60-90 Triangle (Pythagorean)
c = 2a
b = a√3 30 2a
√3 a 90 60 a

7. Midpoint xavg = (x1 + x2)/2 yavg = (y1 + y2)/2 (x2, y2) x1 + x2 2
y1 + y2 2
( , )
(x1, y1)

8. Slope Formula The slope from (x1, y1) to (x2, y2): y2 - y1 ----------
x2 - x1
(x2, y2)
y2 - y1
(x1, y1)
x2 - x1

9. Collinear Points
Points are collinear when they all have the same slope
--or--
Points are collinear when they all fall on the same line
Collinear
Non-Collinear

10. Random Fact!
If the x coordinates between two points on the graph y = x2 add up to 3, then the slope between those two points equals 3.
Let’s say: P1 = (a, a2) and P2 = (3-a, (3-a)2)
(3-a)2 – a2
(3-a) - a
9 – 6a + a2 – a2
3 – a – a
9 – 6a
3 – 2a
3 – 2a = = = =
3 (slope)

11. Distance Formula Using Pythagorean theorem:
(x2 – x1)2 + (y2 – y1)2 = length2
length
y2 - y1
x2 - x1

12. Distance Formula in 3D (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2 = length2

13. Circle Formula
At midpoint (a, b):
(x – a)2 + (y – b)2 = r2
(a, b) r

14. Sphere Formula At midpoint (a, b, c):
(x – a)2 + (y – b)2 + (z – c)2 = r2
(a, b, c) r

15. Parallel and Perpendicular
Parallel lines will have the same slope
Perpendicular lines:
if one line has slope m
the other line has slope -1/m
Slope: -1/m
Slope: m

16. Movement Movement in 2D is symbolized by <a, b>
where a is the change in x
where b is the change in y
Movement in 3D is symbolized by <a, b, c>
where c is the change in z

17. Slope (fun facts) Since slope is: Slope can also be:
∆ means “change in”, so slope can also be:
In terms of movement <x, y>, the slope is simply:
y2 - y1
x2 - x1
change in y
change in x ∆y
----- ∆x y -- x

18. Perpendicular Movements in 3D
Two movements are perpendicular if:
ax + by + cz = 0
Movement 1 is <a, b, c>
Movement 2 is <x, y, z>
<a, b, c>
<x, y, z>

19. Movements Perpendicular to Plane
Ax + By + Cz = D
A, B, C, D are constant
<x, y, z> is always perpendicular to <A, B, C>
<x, y, z> is only moving on one plane
<A, B, C>
<x, y, z>

20. Convex / Concave Polygons
Convex: none of the angles > 180
Concave: 1 or more angles > 180
Convex Polygon
Concave Polygon

21. Sum of all Interior Angles
n is number of sides
360
540
180

22. Number of Diagonals in a Polygon
n(n – 3) 2
n is number of sides 2 5

23. Regular Polygon
All sides and angles are the same 60 a a 60 60 a

24. Perimeter and Area Perimeter of Circle: 2πr
Perimeter of Polygon: Sum of all sides
Area of Circle: πr2
Area of Rectangle: width x height
Area of Triangle: ½ base x height
Area of Trapezoid: ½ (base1 + base2) x height
Area of a Regular Polygon: n(s2/4)tan((n-2)180/2n)
Area of a Cone’s side: πr (height of slant)

25. Prism Surface Area Area = (2 * base area) + (base perimeter * height)
Area = 2πr 2+ 2πrh h

26. Prism Volume
Volume = base area * height r
Volume = πr 2h h

27. Pyramid/Cone Surface Area
Area = base area + sum of side area(s)
∑ means “sum of”
l2 = r2 + h2
Area = πr 2+ πrl l h r

28. Pyramid/Cone Volume Volume = (1/3) base area * height
Volume = (1/3) πr 2h h r

29. Sphere Surface Area and Volume
Area = 4πr2
Volume = 4/3 πr3 r

30. Similar Figures
Have equal angles and a constant ratio between their sides
R = Ratio between their side lengths
R = Ratio between their perimeters
R2 = Ratio between their areas
Side ratio: 2:1
Perimeter ratio: 2:1
Area ratio: 4:1

31. Similar Figures in 3D R = Ratio between their side lengths
R = Ratio between their perimeters
R2 = Ratio between their areas
R3 = Ratio between their volumes
Side ratio: 2:1
Perimeter ratio: 2:1
Area ratio: 4:1
Volume ratio: 8:1

32. Inscribed Angle Angle that touches side
Arc Angle – perimeter which inscribed angle intersects
Arc Angle
Inscribed Angle

33. Inscribed Angle
The measure of an inscribed angle that touches the side is:
Arc Length/2
Arc Length: π
Inscribed Angle: π/2

34. Tangent Lines A tangent line hits only one point of a curve
Point of tangency

35. Tangent Lines Imply Equal Lengths
Two intersecting tangent lines have equal lengths from their points of tangency.

36. Chords
A line connecting two points of a circle

37. Intersecting chords
AB * BD = AC * CD D A B C

38. Secant Lines
Secant lines intersect two points of a curve

39. Intersecting Secant Lines= AB AC AE AD D B E A C