Derivation of Formula

Derivation of Product of First n Terms of Geometric Progression

The product of the first $n$ terms of a Geometric Progression is given by the following:

Given the first term $a_1$ and last term $a_n$:

$P_n = \sqrt{(a_1 \times a_n)^n}$


Given the first term $a_1$ and the common ratio $r$

$P_n = {a_1}^n \times r^{n(n - 1)/2}$


Unit Weights and Densities of Soil

Symbols and Notations
γ, γm = Unit weight, bulk unit weight, moist unit weight
γd = Dry unit weight
γsat = Saturated unit weight
γb, γ' = Buoyant unit weight or effective unit weight
γs = Unit weight of solids
γw = Unit weight of water (equal to 9810 N/m3)
W = Total weight of soil
Ws = Weight of solid particles
Ww = Weight of water
V = Volume of soil
Vs = Volume of solid particles
Vv = Volume of voids
Vw = Volume of water
S = Degree of saturation
w = Water content or moisture content
G = Specific gravity of solid particles

Length of Arc in Polar Plane | Applications of Integration

The length of arc in polar plane is given by the formula:

$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$




The formula above is derived in two ways.

Length of Arc in XY-Plane | Applications of Integration

Arc Length in xy-Plane | Derivation of Formulas | Integral Calculus

The length of arc in rectangular coordinates is given by the following formulas:

$\displaystyle s = \int_{x_1}^{x_2} \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$   and   $\displaystyle s = \int_{y_1}^{y_2} \sqrt{1 + \left(\dfrac{dx}{dy} \right)^2} \, dy$




See the derivations here:

The Three-Moment Equation

The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.

Consider three points on the beam loaded as shown.



Quadrilateral Circumscribing a Circle

Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.

Tangential Quadrilateral



$A = rs$

Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2

Derivation for area

Derivation of Heron's / Hero's Formula for Area of Triangle

For a triangle of given three sides, say a, b, and c, the formula for the area is given by

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


where s is the semi perimeter equal to P/2 = (a + b + c)/2.

Derivation / Proof of Ptolemy's Theorem for Cyclic Quadrilateral

Ptolemy's theorem for cyclic quadrilateral states that the product of the diagonals is equal to the sum of the products of opposite sides. From the figure below, Ptolemy's theorem can be written as

$d_1 d_2 = ac + bd$


Derivation of Formula for Area of Cyclic Quadrilateral

For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by

$A = \sqrt{(s - a)(s - b)(s - c)(s - d)}$


Where s = (a + b + c + d)/2 known as the semi-perimeter.

Derivation of Formula for Lateral Area of Frustum of a Right Circular Cone

The lateral area of frustum of a right circular cone is given by the formula

$A = \pi (R + r) L$


R = radius of the lower base
r = radius of the upper base
L = length of lateral side


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