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Math Formulas

Math is the most commonly used subject in our daily lives, yet it is considered one of the most challenging ones. To make math easier and more practical, we use formulas to simplify calculations. Math without formulas is impossible. In this article, we will explore math formulas and learn how to apply them effectively.

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What are Math Formulas?

Math is incomplete without its formulas. Formulas in mathematics form the foundation for solving real-world problems as well as complex scientific and academic problems. The concepts of mathematics and their application depend extensively on the use of formulas. Mathematical formulas make solving equations and problems easier, as they provide a straightforward method and a structured approach to problems and their solutions.

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Importance of Math Formulas in Problem-Solving

Math formulas are essential tools that we use to solve problems faster and accurately. They help simplify problems that would otherwise seem impossible to solve. We use formulas in various fields. Here are some reasons why math formulas are very important:

 

  • Breaks down complex problems into much simpler problems.

 

  • Formulas guarantee accurate answers.

 

  • We save a lot of time using formulas. By applying the correct formulas, we can solve problems much faster.

 

  • Math formulas are also used in various fields like physics, engineering, robotics even in medicine. 
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History of Math Formulas

Around 2500 BC, Egyptians used formulas to measure land and build pyramids. Much later, mathematicians like Pythagoras introduced formulas for geometry. Another Greek mathematician Euclid, known as the father of Geometry, gave a set of principles called Euclid’s axioms. 

 

Soon after, Indian mathematicians like Brahmagupta and Aryabhatta created formulas for topics like algebra and trigonometry. Arabic scholars like Al-Khwarizmi advanced algorithms and algebraic methods. During the 17th century, scholars like Isaac Newton and Gottfried Leibniz developed formulas for calculus which we use to this day. Today, we continue to use formulas for various fields like physics, engineering, and robotics, making math formulas a vital part of technological progress.

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Major Categories of Math Formulas

We use formulas to make solving problems much easier and also to get accurate results. Here are some of the major categories of math formulas:

 

  • Arithmetic Formulas

 

  • Algebra Formulas

 

  • Geometry Formulas

 

  • Trigonometry Formulas

 

  • Calculus Formulas

 

 

 

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Arithmetic Formulas

Arithmetic formulas are basic mathematical operations like addition, subtraction, multiplication, and division. 

 

The arithmetic operations are: 


Addition (+): a + b = c


Subtraction (-): a - b = c


Multiplication (×): a × b = c


Division (÷): a ÷ b = c or a/b = c

 

Let’s look at a few examples of arithmetic operations using these math formulas

 

Example 1: Jack has 4 Pokemon cards and you have 6 cards. If Jack decides to give his cards to you, how many cards will you have? 
 

Solution: a + b = c


    4 + 6 = 10 cards


So you will have a total of 10 Pokemon cards.
 

Explanation: We use addition because we are combining two amounts. Therefore, 6 cards become 10 cards.

 

 

Example 2: Now Jack takes 2 cards back from your 10 cards. How many cards will you have left?
 

Solution: a - b = c


        10 - 2 = 8 cards


So now you will have a total of 8 cards.
 

Explanation: Jack took two of his cards back from your 10 cards. You will be left with only 8 cards.

 

 

Example 3: Jack now surprised you with 3 unopened packs of pokemon cards each with 5 cards inside. How many cards in total will you get from these packs?
 

Solution: a × b = c


       3 × 5 = 15


You will get a total of 15 cards from the 3 unopened pokemon packs.
 

Explanation: We use multiplication when we have groups of equal size. Adding would take more time, so we use multiplication instead. 

 

 

Example 4:  If you have a total of 30 cards, and you decide to split the cards equally between you and Jack. How many cards would each of you get?
 

Solution: a/b = c


        30/2 = 15


Jack and you would get 15 cards each. 
 

Explanation: Division helps us split things into equal parts. It's especially useful when we want to share or distribute something in equal amounts.
 

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Algebra Formulas

Algebra formulas help solve problems involving unknown variables, making them especially useful when some information is missing – for example, when solving for X.

 

Some important algebraic formulas to remember are: 

 

  • (a + b)2 = a2 + 2ab + b2

 

  • (a - b)2 = a2 - 2ab + b2

 

  • a2 - b2 = (a + b)(a - b)

 

  • a2 + b2 = (a - b)2 + 2ab

 

  • a3 + b3 = (a + b)(a2 - ab +b2)

 

  • a3 - b3 = (a - b)(a2 + ab +b2)

 

  • (a + b)3 = a3 + 3a2b = 3ab2 + b3

 

  • (a - b)3 = a3 - 3a2b = 3ab2 - b3

 

Quadratic formula: ax2 + bx + c = 0
 

Quadratic equation: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Note: a, b, c are coefficients of real numbers where a ≠ 0

 

Example 1: Solve 2x2 - x - 1 = 0, using quadratic formula.
Solution: ax2 + bx + c = 0


Step 1: Find x


    a = 2, b = -1, c = -1


    x = [−b ± √(b² − 4ac)] / (2a)


    x = [−(-1) ± √((-1)² − 4(2)(-1))] / (2(1))


    x = [1 ± √1 + 8] / (4)


    x = [1 ± √9] / 4


Step 2: Solve for both roots (+ and -)


     √b- 4ac = √9 = 3


Now we solve for x:


Add first (for the 1st root):


x = (1 + 3) / 4 = 4/4 = 1 


Subtract next (for the 2nd root):


    x = (1 - 3)/4= -2/4 = -1/2


The roots are: x = 1 or x = -12

 

 

 

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Geometry Formulas

We use geometry to calculate the size or even the space of objects and shapes. Some formulas include area, volume, and perimeter.

 

Some major formulas of geometry are:

 

Perimeter Formulas:

 

  • Square: P = 4s (where s is the side of the square)

 

  • Rectangle: P = 2(l + b) 

 

  • Triangle: P = (a + b + c)

 

  • Circle: C = 2πr (where r is the radius)

 

 

Area Formulas:

 

  • Square: A = s2

 

  • Rectangle: A = L × B

 

  • Triangle: A = 1/2bh (where b is the base and h is the height of the triangle)

 

  • Circle: A = πr2

 

 

Volume formulas:

 

  • Cube: V = s3

 

  • Cuboid: V = L x B x H

 

  • Cylinder: V = πr2h

 

  • Cone: V = 1/3πr2h

 

  • Sphere: V = 4/3πr3

 

Some few examples using these math formulas: 

 

Example 1: Your garden is 6 meters long and 4 meters wide. What is the area of the garden?


Solution: To find area we use the formula: A = l × b


        A =  6 × 4 = 24 sqm

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Trigonometry Formulas

We use trigonometry to understand relationships between angles and sides of triangles, mainly right angles.
The main formulas in trigonometry are:

 

  • Sine (sin): sinθ = Opposite/Hypotenuse

 

  • Cosine (cos): cosθ = Adjacent/Hypotenuse

 

  • Tangent: tanθ = Opposite/Adjacent

 

  • Secant: secθ = Hypotenuse/Adjacent

 

  • Cosecant: cosecc θ = Hypotenuse/Opposite

 

  • Cotangent: cot θ = Adjacent/Opposite

 

Angles from 0° - 360° each have a special value which we use to solve problems.

 

Example 1:  You are standing 30 meters away from the base of a tree. The angle of elevation to the top of the tree is 45°. Find the height of the tree. 
 

Solution: Base of the tree is the adjacent side =  30m
We are trying to find the height of the tree which is the opposite side.

So we will use tanθ because we have the adjacent and the angle.


Angle = 45° = 1 (Tan 45° = 1)


tanθ = Opposite/Adjacent


tan(45°) = h/30


1 = h/30


h = 30m


After solving, we know that the height of the tree is 30 meters. 
 

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Calculus Formulas

Calculus is used to deal with problems that involve change, like speed, growth or even decay. It helps understand things that change over time. 

 

Differentiation: When we want to find how fast something is changing at a given moment in time, we use differentiation.

Formula: d/dt (distance) = speed.


 

Integration: This helps us to calculate the total amount of area or distance during change of time.

Formula: ∫ v(t) dt where v(t) is the velocity function

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Probability and Statistics Formulas

When we want to make predictions or analyze data, we use probability and statistics formulas. Probability tells us how likely an event is, and statistics helps us understand a collection of data.

Some important formulas to remember are:

 

  • Mean = Sum of given data / Total number of data

     
  • Median: For even numbers = sum of the middle two numbers/2 
    For odd numbers = The middle number is the median.

     
  • Standard Deviation =√ ∑(xi - μ)2/n

     
  • Variance = ∑(xi -  x )2/n

Where,


xi = the number in a list of numbers and i is the position of the number, in the data set.


x = mean or average


= Sum of all the terms


(xi - μ)2 = Squared difference


n = total number of data in the data set
 

 

  • Probability P(n) = number of ways n can occur/total number of possible outcomes
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Basic Rules and Properties of Math Formulas

There are some rules and properties we follow when using math formulas:

 

  • Associative Law: This law says that, grouping numbers different when adding or multiplying will not change the result.
    Example: (2 + 3) + 4 = 9 and 2 + (3 + 4) = 9

 

  • Commutative Law: This rule says that you can swap the numbers around when adding or multiplying and the result won't change.
    Example: 2 × 4 = 8 and 4 × 2 = 8

 

  • Distributive Law: This rule helps simplify big problems, by multiplying each part separately, then we add the results together.
    Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 1

 

  • Rules of Power and Roots: Power is a shortcut to multiply the same number many times.
    Example: 23 = 2 × 2 × 2 = 8.
    Roots are the opposite of the power. It helps find the original number when you know its power. Example: Square root of 9 is 3, 3 × 3 = 9



Begin your journey into Math Formulas by exploring key concepts. Understand important math formulas topics in detail by selecting from the list below:

Distance Formula Slope Formula
Recursive Formula Euclidean Distance Formula
Trigonometry Formulas Difference of Squares Formula
Percent Composition Formula Circle Formulas
30-60-90 Triangle Formula Surface Area and Volume Formulas
Population Mean Formula Mean Median Mode Formula
Fahrenheit Formula Conic Sections Formulas
Frequency Distribution Formula Population Change Formula
Double Time Formula Supplementary Angles Formula
SAS Triangle Formula 3D Geometry Formulas
Future Value Simple Interest Formula All Circle Formulas
Dimensional Formula Prime Factorization Formula
Algebraic Sequence Formula  
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Tips and Tricks to Learn Math Formulas

Math can get tricky, but with a few simple tips and tricks, you can make solving problems faster. Here are a few strategies to guide you through:

 

  • Making Use of Mnemonics: Try to create catchy phrases to remember formulas. A very popular Mnemonic is “SOH - CAH - TOA” which stands for sine, cosine, and tangent. You can even make your own mnemonics to make learning much easier.
     

 

  • Visualization Techniques: Draw various diagrams, charts, or graphs to better understand how formulas work. This is especially helpful for subjects like geometry and trigonometry.
     

 

  • Flashcards Can Be a Great Help: Use flashcards to remember formulas, helpful for quick reviews.

     
  • Group Formulas by Topic: Remember, formulas for area are similar, the formulas usually based on the multiplication of base length and height. 

 

  • Break Down Long Formulas: Large formulas can feel overwhelming. Split them into smaller chunks.
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Common Mistakes and How to Avoid Them in Applying Formulas

While working with math formulas, it is common to commit certain mistakes. In this section, we will discuss some common mistakes and how to avoid them.

Mistake 1

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Variables are misinterpreted

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Students may mix up variables or use incorrect values in formulas. Always label diagrams clearly, and always double-check that the correct values are used for each variable.

Mistake 2

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Forgetting to mention the units in measurements.

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If you ignore units, it can lead to incorrect or meaningless answers. Always make sure to write units during calculations.

Mistake 3

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Not following BODMAS rule

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When solving formulas with multiple operations. Make sure to follow the BODMAS rule. Start by solving brackets and powers first, then division, multiplication, and subtraction in that order.

Mistake 4

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Forgetting to place positive or negative signs

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Always be careful with signs. Double check that you’re using the correct signs when solving. 
Example: (-2) + (-3) = -5 (some may confuse the signs and answer it as -1).

Mistake 5

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Rounding answers early can lead to incorrect answers 
 

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Don't round answers in the middle of calculations. Round only in the final step to avoid mistakes.

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Real-World Applications of Math Formulas

Math formulas are used in our daily lives in various different aspects. In this section, we will discuss some real life applications of math formulas.

 

Engineering: Engineers constantly use formulas to design structures like buildings. They calculate the area and volume of the materials to see if it can handle weight safely.


 

Finance: Bankers and investors use math formulas like compound interest to manage loans.


 

Physics: Scientists use math formulas to measure how fast something moves or how long it takes to travel from point A to point B.

 

Trade: Basic trading in all business use math formulas too. The formulas for multiplication, discount, profit and loss are used frequently.

 

Surveys: Median, mean and mode formulas are used when analyzing data collected through surveys and opinion polls.


 

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Solved Examples on Math Formulas

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Problem 1

A teacher collects test scores from her class. The scores are: [ 10, 20, 30, 40, 50]. Calculate the mean, median and variance for these scores.

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Variance = 1000/5 =200.
 

Explanation

First let us calculate the mean.
    
Mean = (10+ 20 + 30 + 40 + 50) 

                               5

= 150 ÷ 5

= 30  

So the mean score or the average marks of the class is 30.

 

Median: Median is the middle number first, let's arrange in ascending order 
    

10, 20, 30, 40, 50.

The middle number is the 3rd number, which is 30. Therefore, the median is 30.
 

Variance: We first find the mean, which we already know is 30.

First subtract the mean from each score, square the result, and then we find the average of these squared differences.

After we subtract the mean from each score we get:
        

Variance = (400 + 100 + 0 + 400 + 100)

                                       5

Variance = 1000 ÷ 5 = 200.

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Problem 2

Solve 16x^2 - 25y^2

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(4x - 5y)(4x + 5y)

Explanation

Step 1: Determine the identity, in this case we use

a2 - b2 =  (a + b)(a - b)


Step 2: Rewrite each term as a square

16x2 = (4x)2

25y2  = (5y)2


Step 3: We now apply the formula

(4x - 5y)(4x + 5y)

So a = 4x and b = 5y

Therefore, the factored form is (4x - 5y)(4x + 5y).
 

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Problem 3

The sides of a triangle are 6cms, 7cms and 8cms. Find the perimeter.

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P = (6 + 7 + 8) = 21 meters.

Explanation

The formula for perimeter of a triangle is P = (a +b + c)

P = (6 + 7 + 8) = 21 meters.
 

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Problem 4

A ladder leans against a vertical wall, reaching a height of 10m. The ladder makes an angle of 60° with the ground. Find the length of the ladder.

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Ladder is the hypotenuse = x (to be found)


Height of the wall is opposite = 10m


Angle = 60°


We will use the sine function because we are dealing with a right-angled triangle.


Formula: sinθ = \(Opposite side\over hypotenuse\)
   

Substitute the values:


sin(60°) = 10x


We know that sin(60°) = √3/2 ≈ 0.866


0.866 = 10/x


x = \(10\over 0.866\)

 

= 11.55 m.

Explanation


So the length of the ladder is 11.55 meters. Using the sine formula, we found the hypotenuse, which is the height of the ladder.


 

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Problem 5

We cut a pizza in 12 slices. If we want to share equally with 3 people, how much would each person get?

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12 ÷ 4 = 3
 

Explanation

Each person gets 3 slices each. We use division to split the total equally with all three members.

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FAQs on Math Formulas

1.How to calculate the area of a circle?

Area of circle = πr2

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2. What is the full form of BODMAS?

Brackets, Order (that is powers or roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

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3.How to calculate probability?

Divide the number of favorable events by the total number of possible events.

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4. Who is the father of Statistics?

Sir Ronald Aylmer Fisher is the father of statistics.

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5.When was trigonometry first discovered?

It was discovered around 120 BC by the Greek mathematician Hipparchus.

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6.What are the basic operations of math?

The basic operations applied in mathematics are addition, multiplication, subtraction, and division.

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7.What are the four types of math formulas?

Arithmetic formulas: These deal with basic operations, numbers, and shapes.

 


Algebraic formulas: These are used for expressions, equations, and polynomials.

 


Geometric formulas: These involve shapes, sizes, and properties of figures in 2D and 3D.

 


Trigonometric formulas: These deal with angles, triangles, and trigonometric identities.

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8.What is the BODMAS rule?

The term BODMAS stands for Bracket, Order, Division, Multiplication, Addition, and Subtraction. According to this rule, while solving an equation, the mathematical operations should be conducted in the order BODMAS.

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9.What is this () called in math?

They are called parentheses or round brackets in mathematics.
 

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10.What is {} called in terms of math?

{ } are called curly brackets or braces.

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Explore More Math Topics

From Numbers to Geometry and beyond, you can explore all the important Math topics by selecting from the list below:
 

Numbers Multiplication Tables
Geometry Algebra
Calculus Measurement
Trigonometry Commercial Math
Data Math Questions
Math Calculators Math Worksheets
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