Basic Math Formulas from 5th Grade to B.Tech and Diploma

Mathematics forms the backbone of many disciplines, from the early years of schooling to advanced fields like engineering and technology. Understanding and mastering basic math formulas lay the groundwork for success in more complex mathematical concepts. Whether you’re a student starting from elementary school or pursuing a diploma or bachelor’s degree in engineering, the journey begins with mastering these fundamental formulas.

Essential Math Formulas for 5th Grade

Example Use: To illustrate, if you have a square with a side length of 5 units, the perimeter (P) can be calculated as: P=4×5=20 unitsP = 4 \times 5 = 20 \text{ units}P=4×5=20 units

Essential Math Formulas for 6th Grade

Example Use: To apply these formulas, consider calculating the area of a rectangle with a length of 8 units and width of 5 units: A=8×5=40 square unitsA = 8 \times 5 = 40 \text{ square units}A=8×5=40 square units

Essential Math Formulas for 7th Grade

Example Use: To apply these formulas, consider finding the circumference of a circle with a radius of 6 units: C=2π×6=12π unitsC = 2 \pi \times 6 = 12 \pi \text{ units}C=2π×6=12π units

[Also Read: MBBS Full Form: A Comprehensive Detailed Guide About MBBS]

Essential Math Formulas for 8th Grade

Example Use: To apply these formulas, consider calculating the area of a rectangle with a length of 12 units and width of 8 units: A=12×8=96 square unitsA = 12 \times 8 = 96 \text{ square units}A=12×8=96 square units

Essential Math Formulas for 9th Grade

Example Use: To apply these formulas, consider solving a quadratic equation for xxx: 2×2+5x−3=02x^2 + 5x – 3 = 02×2+5x−3=0

Using the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​ where a=2a = 2a=2, b=5b = 5b=5, and c=−3c = -3c=−3: x=−5±52−4⋅2⋅(−3)2⋅2x = \frac{-5 \pm \sqrt{5^2 – 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}x=2⋅2−5±52−4⋅2⋅(−3)​​ x=−5±25+244x = \frac{-5 \pm \sqrt{25 + 24}}{4}x=4−5±25+24​​ x=−5±494x = \frac{-5 \pm \sqrt{49}}{4}x=4−5±49​​ x=−5±74x = \frac{-5 \pm 7}{4}x=4−5±7​

So, the solutions are x=24=0.5x = \frac{2}{4} = 0.5x=42​=0.5 and x=−124=−3x = \frac{-12}{4} = -3x=4−12​=−3.

Essential Math Formulas for 10th Grade

Example Use: To apply these formulas, consider finding the volume of a rectangular prism with dimensions:

• Length = 5 units
• Width = 3 units
• Height = 4 units

V=Base Area×HeightV = \text{Base Area} \times \text{Height}V=Base Area×Height Base Area=5×3=15 square units\text{Base Area} = 5 \times 3 = 15 \text{ square units}Base Area=5×3=15 square units V=15×4=60 cubic unitsV = 15 \times 4 = 60 \text{ cubic units}V=15×4=60 cubic units

[Also Read: NET Full Form: A Comprehensive Detailed Guide About NET]

Essential Math Formulas for 11th Grade

Example Use: To apply these formulas, consider finding the volume of a sphere with a radius of 6 units: V=43π×63V = \frac{4}{3} \pi \times 6^3V=34​π×63 V=43π×216V = \frac{4}{3} \pi \times 216V=34​π×216 V=288π cubic unitsV = 288 \pi \text{ cubic units}V=288π cubic units

Essential Math Formulas for 12th Grade

Example Use: To apply these formulas, consider finding the limit of a function f(x)=x2−1x−1f(x) = \frac{x^2 – 1}{x – 1}f(x)=x−1×2−1​ as xxx approaches 1: lim⁡x→1×2−1x−1\lim_{x \to 1} \frac{x^2 – 1}{x – 1}limx→1​x−1×2−1​

Direct substitution gives 12−11−1=00\frac{1^2 – 1}{1 – 1} = \frac{0}{0}1−112−1​=00​, which is indeterminate. Applying algebraic manipulation or L’Hôpital’s rule would resolve this limit.

Essential Math Formulas for B.Tech

Example Use: To apply these formulas, consider solving a second-order linear differential equation: d2ydx2+3dydx+2y=0\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0dx2d2y​+3dxdy​+2y=0

The characteristic equation is r2+3r+2=0r^2 + 3r + 2 = 0r2+3r+2=0, which factors to (r+1)(r+2)=0(r + 1)(r + 2) = 0(r+1)(r+2)=0. Thus, the roots are r=−1r = -1r=−1 and r=−2r = -2r=−2, leading to the general solution: y(x)=C1e−x+C2e−2xy(x) = C_1 e^{-x} + C_2 e^{-2x}y(x)=C1​e−x+C2​e−2x

B.Tech 1st Year

Math Formulas for B.Tech 1st Year

1. Calculus
   • Derivative of a function: ddx[f(x)]\frac{d}{dx}[f(x)]dxd​[f(x)]
   • Integration: ∫f(x) dx\int f(x) \, dx∫f(x)dx
   • Fundamental Theorem of Calculus: ∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) – F(a)∫ab​f(x)dx=F(b)−F(a)
2. Linear Algebra
   • Matrix Operations: Addition, Subtraction, Scalar Multiplication
   • Determinant of a Matrix: det⁡(A)\det(\mathbf{A})det(A)
   • Inverse of a Matrix: A−1\mathbf{A}^{-1}A−1
3. Differential Equations
   • First-Order Differential Equations: dydx=f(x,y)\frac{dy}{dx} = f(x, y)dxdy​=f(x,y)
   • Second-Order Differential Equations: d2ydx2=f(x,y,dydx)\frac{d^2y}{dx^2} = f(x, y, \frac{dy}{dx})dx2d2y​=f(x,y,dxdy​)
4. Trigonometry
   • Trigonometric Identities: sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1sin2(θ)+cos2(θ)=1
   • Trigonometric Functions: sin⁡(θ),cos⁡(θ),tan⁡(θ)\sin(\theta), \cos(\theta), \tan(\theta)sin(θ),cos(θ),tan(θ)
5. Complex Numbers
   • Basic Operations: Addition, Subtraction, Multiplication, Division
   • Polar Form: z=r(cos⁡θ+isin⁡θ)z = r(\cos \theta + i \sin \theta)z=r(cosθ+isinθ)

B.Tech 2nd Year

Math Formulas for B.Tech 2nd Year

1. Vector Calculus
   • Gradient: ∇f=(∂f∂x,∂f∂y,∂f∂z)\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)∇f=(∂x∂f​,∂y∂f​,∂z∂f​)
   • Divergence: ∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}∇⋅F=∂x∂Fx​​+∂y∂Fy​​+∂z∂Fz​​
   • Curl: ∇×F=(∂Fz∂y−∂Fy∂z,∂Fx∂z−∂Fz∂x,∂Fy∂x−∂Fx∂y)\nabla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} – \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} – \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} – \frac{\partial F_x}{\partial y} \right)∇×F=(∂y∂Fz​​−∂z∂Fy​​,∂z∂Fx​​−∂x∂Fz​​,∂x∂Fy​​−∂y∂Fx​​)
2. Probability and Statistics
   • Probability Distribution Functions: Normal, Binomial, Poisson
   • Expected Value: E(X)E(X)E(X)
   • Variance: Var(X)\text{Var}(X)Var(X)
3. Numerical Methods
   • Numerical Integration: Trapezoidal Rule, Simpson’s Rule
   • Numerical Differentiation: Forward Difference, Central Difference
4. Partial Differential Equations
   • Heat Equation: ∂u∂t=α∂2u∂x2\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}∂t∂u​=α∂x2∂2u​
   • Wave Equation: ∂2u∂t2=c2∂2u∂x2\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}∂t2∂2u​=c2∂x2∂2u​

B.Tech 3rd Year

Math Formulas for B.Tech 3rd Year

1. Advanced Engineering Mathematics
   • Fourier Series: f(x)=a02+∑n=1∞(ancos⁡nπxL+bnsin⁡nπxL)f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n\pi x}{L} + b_n \sin \frac{n\pi x}{L} \right)f(x)=2a0​​+∑n=1∞​(an​cosLnπx​+bn​sinLnπx​)
   • Laplace Transform: L{f(t)}=∫0∞e−stf(t) dt\mathcal{L}\{f(t)\} = \int_0^{\infty} e^{-st} f(t) \, dtL{f(t)}=∫0∞​e−stf(t)dt
   • Z-Transform: X(z)=∑n=0∞x(n)z−nX(z) = \sum_{n=0}^{\infty} x(n) z^{-n}X(z)=∑n=0∞​x(n)z−n
2. Control Systems
   • Transfer Function: G(s)=Y(s)X(s)G(s) = \frac{Y(s)}{X(s)}G(s)=X(s)Y(s)​
   • Bode Plot: Magnitude and Phase plots
3. Numerical Analysis
   • Root Finding Methods: Newton-Raphson Method, Bisection Method
   • Optimization Techniques: Gradient Descent, Newton’s Method

B.Tech 4th Year

Math Formulas for B.Tech 4th Year

1. Advanced Topics in Mathematics
   • Complex Analysis: Residue Theorem, Analytic Functions
   • Advanced Differential Equations: Nonlinear Systems, Stability Analysis
2. Mathematical Modeling
   • Modeling Techniques: Differential Equations, Optimization Models
   • Simulation Methods: Monte Carlo Simulation, Discrete Event Simulation
3. Advanced Probability and Statistics
   • Multivariate Statistics: Covariance Matrix, Correlation Coefficient
   • Time Series Analysis: Autocorrelation Function, Stationarity
4. Advanced Numerical Methods
   • Finite Element Method (FEM): Discretization, Solving PDEs
   • Computational Fluid Dynamics (CFD): Navier-Stokes Equations, Boundary Conditions

Example Use: To apply these formulas, consider solving a second-order differential equation from B.Tech 1st year: d2ydx2+4dydx+3y=0\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 0dx2d2y​+4dxdy​+3y=0

The characteristic equation is r2+4r+3=0r^2 + 4r + 3 = 0r2+4r+3=0, which factors to (r+3)(r+1)=0(r + 3)(r + 1) = 0(r+3)(r+1)=0. Thus, the roots are r=−3r = -3r=−3 and r=−1r = -1r=−1, leading to the general solution: y(x)=C1e−3x+C2e−xy(x) = C_1 e^{-3x} + C_2 e^{-x}y(x)=C1​e−3x+C2​e−x

Essential Math Formulas for Diploma Programs

Example Use: To apply these formulas, consider calculating the perimeter of a rectangle with a length of 8 units and a width of 5 units: Perimeter=2×(8+5)=26 units\text{Perimeter} = 2 \times (8 + 5) = 26 \text{ units}Perimeter=2×(8+5)=26 units

Essential Math Formulas for Diploma Programs

1. Basic Algebra
   • Linear Equations: ax+by=cax + by = cax+by=c
   • Quadratic Equations: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0
   • Factoring: a2−b2=(a+b)(a−b)a^2 – b^2 = (a+b)(a-b)a2−b2=(a+b)(a−b)
2. Geometry
   • Area of Rectangle: Area=length×width\text{Area} = \text{length} \times \text{width}Area=length×width
   • Area of Triangle: Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}Area=21​×base×height
   • Perimeter: Perimeter=2×(length+width)\text{Perimeter} = 2 \times (\text{length} + \text{width})Perimeter=2×(length+width)
3. Trigonometry
   • Sine Function: sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}sin(θ)=HypotenuseOpposite​
   • Cosine Function: cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}cos(θ)=HypotenuseAdjacent​
   • Tangent Function: tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}tan(θ)=AdjacentOpposite​
4. Basic Calculus
   • Derivative: ddx[f(x)]\frac{d}{dx}[f(x)]dxd​[f(x)]
   • Integral: ∫f(x) dx\int f(x) \, dx∫f(x)dx
5. Statistics
   • Mean: xˉ=∑i=1nxin\bar{x} = \frac{\sum_{i=1}^n x_i}{n}xˉ=n∑i=1n​xi​​
   • Standard Deviation: σ=∑i=1n(xi−xˉ)2n\sigma = \sqrt{\frac{\sum_{i=1}^n (x_i – \bar{x})^2}{n}}σ=n∑i=1n​(xi​−xˉ)2​​
6. Financial Mathematics
   • Simple Interest: I=P×r×tI = P \times r \times tI=P×r×t
   • Compound Interest: A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}A=P(1+nr​)nt

Example Use: To apply these formulas, consider finding the area of a triangle with base 6 units and height 4 units: Area=12×6×4=12 square units\text{Area} = \frac{1}{2} \times 6 \times 4 = 12 \text{ square units}Area=21​×6×4=12 square units

Conclusion

Mastering basic math formulas from elementary school to engineering is crucial for building strong problem-solving skills. These formulas lay the foundation for understanding complex concepts in mathematics and their practical applications across various disciplines. Embracing these fundamentals ensures proficiency and confidence in tackling challenges throughout academic and professional journeys.