# 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 $a_n$:

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

## Unit Weights and Densities of Soil

**Symbols and Notations**

γ, γ_{m} = Unit weight, bulk unit weight, moist unit weight

γ_{d} = Dry unit weight

γ_{sat} = Saturated unit weight

γ_{b}, γ' = Buoyant unit weight or effective unit weight

γ_{s} = Unit weight of solids

γ_{w} = Unit weight of water (equal to 9810 N/m^{3})

W = Total weight of soil

W_{s} = Weight of solid particles

W_{w} = Weight of water

V = Volume of soil

V_{s} = Volume of solid particles

V_{v} = Volume of voids

V_{w} = 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 on polar plane is given by the formula:

The formula above is derived in two ways. See it here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-polar-pla...

## Length of Arc in XY-Plane | Applications of Integration

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

See the derivations here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-xy-plane-...

## 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.

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## Quadrilateral Circumscribing a Circle

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

Area,

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

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

## 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

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

where

R = radius of the lower base

r = radius of the upper base

L = length of lateral side