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.

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. 

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.

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
   
• Probability and Statistics Formulas

Arithmetic formulas form the base of all mathematical learning. They cover the four main operations: Addition, Subtraction, Multiplication, and Division, that help students calculate and understand numbers with ease, preparing them for problem-solving in advanced mathematics. Here are the following basic arithmetic operations:  

Addition (+): Addition is the process of combining two or more numbers to find their total or sum.

Formula: a + b = c
Here, a and b are the numbers being added, and c is the resulting sum.

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

Solution:
Number of cards Jack has = 4
Number of cards John has = 6 
After Jack gives John 4 cards, John will have  = a + b = c
= 4 + 6 
= 10      
                
So John will have a total of 10 Pokemon cards with him.

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

Subtraction (-): Subtraction is used to find the difference between two numbers.

Formula: a − b = c
Here, a is the minuend, b is the subtrahend, and c is the difference.

Example : Jack takes 2 cards back from John's 10 cards. How many cards will John have left?

Solution:
Number of Cards John has = 10
Number of cards Jack takes away from John = 2
Number of cards remaining with John = a - b = c 
= 10 - 2
= 8

So now John will have a total of 8 cards with him.

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

Multiplication (×): Multiplication is the process of adding a number to itself a specific number of times.

Formula: \(a \times b = c\)
Here, a is the number being multiplied (the multiplicand), b shows how many times 'a' is multiplied by itself (the multiplier), and c is the final result (the product).

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

Solution:
Jack gives John 3 packs of Pokemon cards.
Each pack contains 5 cards.

To find the total number of cards John receives:
Total cards = \(a \times b = c\)
= Number of packs × Cards per pack
= 3 × 5
= 15

Therefore, John 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. 

Division (÷): Division is the process of splitting a number into equal groups.

Example formula: \(a \over b\) = c  OR  a ÷ b = c 
Here, a (the dividend) is divided by b (the divisor) to get c (the quotient).

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

Solution:
Number of cards John has = 30
Number of people sharing the cards = 2
Number of cards each person gets = \({a \over b} = c\)
= Number of Cards John has \ Total  Number of People
= \({30} \over 2\)
= 15

Jack and John 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.
 

Algebraic expressions are mathematical phrases involving variables like x,y, constants, and operations. They form the basis for solving equations with unknowns. Algebraic formulas provide systematic methods to simplify and manipulate these expressions, enabling precise solutions to linear, quadratic, and polynomial equations

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 Equation: ax² + bx + c = 0

A quadratic equation is a second-degree polynomial equation expressed in the above form, where a, b, and c are constants with a ≠ 0, and x is the unknown variable.

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

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 = \(−b ± \sqrt{(b² − 4ac)} \over {2a}\)

 x =\( {-(-1) \pm \sqrt {-1^2 -4(2) -1 }} \over 2(1)\)

 x = \(1 ± \sqrt {1 - (-8)} \over 4\)

 x = \(1 ± \sqrt 9 \over 4\)

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

 \(  \sqrt {b^2 - 4ac}= \sqrt 9 = 3\)

Now we solve for x:

Add first (for the 1st root):

x = \({1 + 3 \over 4} = {4 \over 4} = 1\)

Subtract next (for the 2nd root):

    x = \({1 - 3 \over 4} = {-2 \over 4} = - {1 \over 2}\)

The roots are: x = 1 or x = \(- {1 \over 2}\)

Geometry is a branch of mathematics that deals with the properties, measurement, and relationships of points, lines, surfaces, and solids. Geometry is used to calculate the length, area, volume, and perimeter of a variety of shapes or objects. These calculations are used in real-life situations such as construction, design, and engineering.

Some major formulas of geometry are:
 

Perimeter Formulas:

Perimeter is the total distance around a two-dimensional shape.
 

• Square:
  
  Add all four sides. Since all sides are equal, multiply one side by 4.
  
  P = 4 × s, where s is the length of the side.

• Rectangle:
  
  Add the lengths of two sides and the two widths, and then multiply by two.
  
  P = 2 (l + b), where l is length and b is breadth. 

• Triangle:
  
  The perimeter of a triangle is the sum of its three sides because it measures the total distance around the shape.
  
  P = (a + b + c), where a, b, c are the lengths of the sides.

• Circle:
  
  Circumference is the distance around a circle based on the radius.
  
  C = 2πr, where r is the radius.
   

Area Formulas:

Area is the measurement of the surface enclosed by a shape, expressed in square units.
 

• Square:
  
  The area is found by multiplying a side by itself because all four sides are equal, covering a square-shaped surface.
  
  A = s², where 's' is the side.
  
   
• Rectangle:
  
  It represents the total number of square units covering the rectangle.
  
  A = L × B, where L is length and B is breadth. 

• Triangle:
  
  A triangle is half of a corresponding rectangle.
  
  A = \({ 1\over 2} \times b \times h\) (where b is the base and h is the height of the triangle)

• Circle:
  
  This gives us the surface enclosed by the circular boundary.
  
  A = πr², where r is the radius.

Volume formulas:

Volume is the measurable amount of three-dimensional space enclosed within an object or shape, expressed in cubic units.

• Cube: V = s³, where 's' is the side.

• Cuboid: V = L x B x H, where L is length, B is breadth and H is height. 

• Cylinder: V = πr²h, where r is radius and h is height. 

• Cone: V = \(1\over3\)πr²h, where r is radius and h is height. 

• Sphere: V = \(4 \over 3\)πr³, where r is radius and h is height. 

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:

Length of garden = 6 meters
Width of garden = 4 meters
To find area we use the formula: A = L × B
Area of garden = 6 × 4 = 24 square meters

The total area of the garden covers 24 square meters.

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles, particularly right-angled triangles.

The main formulas in trigonometry are:
 

• Sine (sin): sin θ = \(Opposite \over Hypotenuse\)

• Cosine (cos): cos θ = \(Adjacent \over Hypotenuse\)

• Tangent: tan θ = \(Opposite \over Adjacent\)

Reciprocal Ratios:

In addition to these, there are three reciprocal ratios that are also useful for advanced problem-solving:
 

• Secant: sec θ = \({1 \over cos \theta } =\) \(Hypotenuse \over Adjacent\)

• Cosecant: cosec θ = \({1 \over sin \theta } =\) \(Hypotenuse \over Opposite\)

• Cotangent: cot θ = \({1 \over tan \theta } =\) \(Adjacent \over Opposite\)

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
                 Angle = 45° = 1 (Tan 45° = 1)

To find the height of the tree, we use the tangent function because tangent relates the angle of elevation to the ratio of the opposite side (height of the tree) over the adjacent side (distance from the tree).

tan θ = \(Opposite \over Adjacent\)

tan(45°) =\(h \over 30\)

1 =\(h \over 30\)
 

h = 30 × 1

h = 30 m

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

Calculus is a fundamental branch of mathematics that studies how quantities change and accumulate. It provides tools to analyze rates of change and total accumulation in various real-world phenomena such as motion, growth, and decay. 

Differentiation: Differentiation allows us to determine the instantaneous rate at which a quantity changes with respect to another variable, often time. It answers questions like "How fast is an object moving at a specific moment?"

Formula: \({d} \over {dt}\) (distance) = speed.

Here,
d stands for a very small (infinitesimal) change or difference in a quantity.
dt represents a very small change in the variable t, often time.

Integration: Integration is the process of finding the total accumulation of a quantity, such as the area under a curve or the total distance traveled over a period of time.

Formula: \(\int v(t) dt\), where v(t) is the velocity function

Here,
v(t) is the velocity function, and dt represents a very small change or interval in time.

The integral sums up the velocity over these tiny time intervals to calculate the total distance covered.

Probability and statistics are fundamental branches of mathematics that enable students to analyze data and measure uncertainty. Probability quantifies the chance of events occurring, while Statistics provides methods to collect, organize, and interpret data for informed decision-making.
 

Some important formulas to remember are:
 

• Mean: The mean represents the central value of a data set, found by dividing the sum of all data points by the total number of points.
  
  Formula = Sum of given data / Total number of data
   
• Median: The median is the middle value of an ordered data set. 
  
  Formulas:
  For Even numbers = Sum of the Middle Two Numbers / 2 
  For Odd numbers = The middle number is the median.
   
• Standard Deviation: Standard deviation is the square root of variance and represents the spread of the data in the original units:
  
  Formula = \({\sqrt { ∑(x_i - μ)^2} } \over {n} \)
  
  Here,
  x_i: Represents each individual data point in the data set.
  μ:  Denotes the mean (average) of the entire population data.
  n: Total number of data points in the sample.
   
• Variance: Variance measures how far each number in the data set is from the mean, on average, squared:
  
  Formula = \( {∑(x_i - \bar{x})^2} \over n\)

Here,
x_i = 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): Probability quantifies the chance that a specific event will happen, expressed as a number between 0 (impossible) and 1 (certain).
  
  Formula = Number of favorable ways n can occur / Total number of possible outcomes

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

• Associative Law: Associative Law states 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: Commutative Law 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: Distributive Law 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:

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:

1. 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.
    
2. Visualization Techniques: Draw various diagrams, charts, or graphs to better understand how formulas work. This is especially helpful for subjects like geometry and trigonometry.
    
3. Using Flashcards: Use flashcards to remember formulas, helpful for quick reviews.
    
4. Group Formulas by Topic: Remember, formulas for area are similar, the formulas usually based on the multiplication of base length and height. 
    
5. Break Down Long Formulas: Large formulas can feel overwhelming. Split them into smaller chunks.

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.
 

1. 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.
    
2. Finance: Bankers and investors use math formulas like compound interest to manage loans.
    
3. Physics: Scientists use math formulas to measure how fast something moves or how long it takes to travel from point A to point B.
    
4. Trade: Basic trading in all business use math formulas too. The formulas for multiplication, discount, profit and loss are used frequently.
    
5. Surveys: Median, mean and mode formulas are used when analyzing data collected through surveys and opinion polls.

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