Triangle

Angle Sum Property

$$A+B+C=180^\circ$$

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Exterior Angle Property

The exterior angle of a triangle is equal to the sum of its two opposite interior angles.

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Area of a Triangle

$$\text{Area}=\frac{1}{2}bh$$

where:

• (b) = base
• (h) = height

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Heron’s Formula

For sides (a), (b), and (c):

$$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$

where

$$s=\frac{a+b+c}{2}$$

is the semi-perimeter.

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Equilateral Triangle

Area

$$\frac{\sqrt{3}}{4}a^2$$

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Height

$$\frac{\sqrt{3}}{2}a$$

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Pythagoras Theorem

$$a^2+b^2=c^2$$

where (c) is the hypotenuse.

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Important Pythagorean Triplets

Memorize:

3, 4, 5

5, 12, 13

8, 15, 17

7, 24, 25

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Circle

Radius:

r

Diameter:

2r

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Circumference

$$2\pi r$$

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Area

$$\pi r^2$$

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Semicircle

Area

$$\frac{1}{2}\pi r^2$$

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Quadrilateral

Angle Sum

$$360^\circ$$

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Polygon

Interior Angle Sum

$$(n-2)\times180^\circ$$

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2. MENSURATION

Rectangle

Area

$$lb$$

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Perimeter

$$2(l+b)$$

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Square

Area

$$a^2$$

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Perimeter

$$4a$$

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Diagonal

$$a\sqrt{2}$$

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Cube

Volume

$$a^3$$

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Surface Area

$$6a^2$$

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Space Diagonal

$$a\sqrt{3}$$

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Cuboid

Volume

$$lbh$$

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Total Surface Area

$$2(lb+bh+hl)$$

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Diagonal

$$\sqrt{l^2+b^2+h^2}$$

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Cylinder

Volume

$$\pi r^2h$$

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Curved Surface Area (CSA)

$$2\pi rh$$

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Total Surface Area (TSA)

$$2\pi r(r+h)$$

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Cone

Volume

$$\frac{1}{3}\pi r^2h$$

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Curved Surface Area

$$\pi rl$$

where (l) is the slant height.

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Sphere

Volume

$$\frac{4}{3}\pi r^3$$

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Surface Area

$$4\pi r^2$$

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3. COORDINATE GEOMETRY

Given points:

$$(x_1,y_1) \quad\text{and}\quad (x_2,y_2)$$

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Distance Formula

$$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

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Midpoint Formula

$$\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$$

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Section Formula

For internal division in the ratio (m:n):

$$\left( \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n} \right)$$

Rare, but important.

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4. SET THEORY & VENN DIAGRAMS

Two Sets

Most important formula:

$$n(A\cup B) = n(A)+n(B)-n(A\cap B)$$

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Three Sets

$$n(A\cup B\cup C) = n(A)+n(B)+n(C) -n(A\cap B) -n(B\cap C) -n(C\cap A) +n(A\cap B\cap C)$$

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Universal Set

Everything under consideration.

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Complement

$$A'=U-A$$

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5. PERMUTATION

Arrangement matters.

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Formula

$${}^nP_r = \frac{n!}{(n-r)!}$$

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Special Case

If all objects are arranged:

$$n!$$

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Circular Permutation

For distinct objects arranged around a circle:

$$(n-1)!$$

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6. COMBINATION

Selection matters.

Order does not matter.

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Formula

$${}^nC_r = \frac{n!}{r!(n-r)!}$$

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Relationship

$${}^nP_r = {}^nC_r\times r!$$

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Important Values

5C1 = 5

5C2 = 10

5C3 = 10

5C4 = 5

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Factorials

Must memorize.

0! = 1

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5040

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7. PROBABILITY

Formula

$$P(E) = \frac{\text{Favourable Outcomes}} {\text{Total Outcomes}}$$

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Range

$$0\le P(E)\le1$$

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Impossible Event

$$P(E)=0$$

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Certain Event

$$P(E)=1$$

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Complement Rule

$$P(E')=1-P(E)$$

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Dice Probability

Single die:

Total outcomes = 6

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Coin Probability

Single coin:

H

T

Total outcomes:

2

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Cards

Important facts:

52 cards

26 red

26 black

13 cards in each suit

4 suits

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8. CLOCKS

Very important.

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Clock Angle Formula

$$\left| \frac{11M}{2}-30H \right|$$

where:

• (H) = hour
• (M) = minutes

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Facts

Hour hand moves:

0.5° per minute

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Minute hand moves:

6° per minute

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Right Angle

$$90^\circ$$

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Straight Angle

$$180^\circ$$

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9. CALENDARS

Leap Year

A year is a leap year if it is:

Divisible by 4

Not divisible by 100

OR divisible by 400

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Examples:

2000 ✔

1900 ✘

2024 ✔

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Odd Days

Normal year:

365 days

1 odd day

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Leap year:

366 days

2 odd days

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Day Cycle

7 days = 0 odd days

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Number of Days in Each Month

Jan – 31

Feb – 28/29

Mar – 31

Apr – 30

May – 31

Jun – 30

Jul – 31

Aug – 31

Sep – 30

Oct – 31

Nov – 30

Dec – 31

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FACTORIALS TO MEMORIZE

0! = 1

1! = 1

2! = 2

3! = 6

4! = 24

5! = 120

6! = 720

7! = 5040

8! = 40320

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MOST IMPORTANT FORMULAS OF SHEET 4

Triangle Area

$$\frac{1}{2}bh$$

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Circle Area

$$\pi r^2$$

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Distance Formula

$$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

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Permutation

$${}^nP_r = \frac{n!}{(n-r)!}$$

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Combination

$${}^nC_r = \frac{n!}{r!(n-r)!}$$

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Probability

$$P(E) = \frac{\text{Favourable Outcomes}} {\text{Total Outcomes}}$$

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Clock Angle Formula

$$\left| \frac{11M}{2}-30H \right|$$

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