Math Internals all equations

1. Area of a triangle

\frac{1}{2} ab \sin C

2. Sine Rule

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

3. a^0

1 (anything to the power 0 = 1)

4. a^m \times a^n

a^{m+n}

5. a^m/a^n

a^{m-n}

6. (a^m)^n

a^{m \times n} powers inside powers means powers multiply

7. 1/a^m

a^{-m}

8. \sqrt[n]{a}

a^{\frac{1}{n}}

9. \sqrt[n]{a^m}

a^{\frac{m}{n}}

10. a^{-\frac{m}{n}}

\sqrt[n]{a^{-m}} = \sqrt[n]{\frac{1}{a^m}}

11. Real Numbers

All numbers on a number line (negative and 0) all the way to infinity

12. Natural Numbers

all numbers from 1 to infinity

13. Integers

whole numbers (negative and positive)

14. Rational numbers

All numbers that can be written as a ratio of integers. E.g: 3.5=\frac{7}{2}, 2.4 = \frac{12}{5}

15. Irrational Numbers

numbers that cannot be written as a ratio of integers. E.g Roots of prime numbers

16. Arc length of a sector

\frac{\theta}{360} \times 2\pi r

17. Area of a sector

Area of a Sector = \left(\frac{\theta}{360}\right) \times \pi r^2

18. Perimeter of a Sector

Arc length + 2r

19. Area of a circle

\pi r^2

20. Perimeter of a circle

2\pi r

21. Surface area of a sphere

4\pi r^2

22. Area of a square

s^2

23. Surface area of a cube

6s^2

24. Volume of a cube

s^3

25. Area of a rectangle

A=lw

26. Surface Area of a rectanglular prism

2lb + 2lw + 2lh

27. Volume of a rectangular prism

L x W x H

28. Area of a parallelogram

A=bh

29. Area of a trapezium

\frac{1}{2}(a+b)h

30. Curved SA of a Cylinder

2\pi rh

31. Total SA of a cylinder

2\pi rh + 2\pi r^2

32. Volume of a Cylinder

V=\pi r^2h

33. Curved SA of a cone

\pi rl

34. Total SA of a cone

\pi rl + \pi r^2

35. Volume of a cone

\frac{1}{3}\pi r^2h (1/3 of the volume of a cylinder)

36. Volume of a Triangular Pyramid

V=\frac{1}{3}(\frac{1}{2}bh)h

37. Volume of a triangular prism

V=(\frac{1}{2}bh)h = Area of a triangle * height

38. Volume of a rectangular pyramid

V=\frac{1}{3}lwh

39. Angle at the Center

The angle at the center of a circle is twice the angle at the circumference, when both angles are subtended by the same arc.

40. Angle in a Semicircle

The angle in a semicircle is always 90 degrees.

41. Angles in the Same Segment

Angles subtended by the same arc (or chord) in the same segment of a circle are equal. (Bowtie)

circumference, when both angles are subtended by the same arc.

42. Cyclic Quadrilateral

The opposite angles of a cyclic quadrilateral (a quadrilateral whose vertices all lie on a circle) add up to 180 degrees.

43. Tangent-Radius Property

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

44. Tangents from a Point

Tangents drawn from the same point to a circle are equal in length.

45. Alternate Segment Theorem

The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

46. Midpoint of a line

\left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right)

47. Gradient of a line

\frac{y2 - y1}{x2 - x1}

Also known as rise over run

48. Equation of a straight line

y = mx + c

m is the gradient

c is the y intercept

49. Equation of a straight line (point gradient form)

y - y1 = m(x - x1)

Given the gradient 'm' and a point (x1, y1)

50. Parallel Lines

Two lines are parallel if their gradients are equal.

m1 = m2

51. Perpendicular Lines

Two lines are perpendicular if the product of their gradients is -1.

m1 \times m2 = -1

52. Distance Between Two Points

The distance d between two points (x1, y1) and (x2, y2) is given by:

\sqrt{(x2 - x1)^2 + (y2 - y1)^2}.

53. Gradient and Angle of Inclination

If a line makes an angle \theta with the positive x-axis, then the gradient m of the line is given by:

m = \tan(\theta).

54. Intercept Form of a Line

The intercept form of a line is given by:

\frac{x}{a} + \frac{y}{b} = 1

where a is the x-intercept and b is the y-intercept.

55. General Form of a Line

For the general equation of a line Ax + By + C = 0, the gradient m and y-intercept c can be found as:

m = -\frac{A}{B}

c = -\frac{C}{B}.