# Derivation of Formula

## Derivation of Formula for Radius of Circumcircle

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by

where A_{t} is the area of the inscribed triangle.

## Derivation of Formula for Radius of Incircle

The radius of incircle is given by the formula

where A_{t} = area of the triangle and s = semi-perimeter.

## Derivation of Cosine Law

The following are the formulas for cosine law for any triangles with sides *a*, *b*, *c* and angles *A*, *B*, *C*, respectively.

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

## Derivation of Sine Law

For any triangles with vertex angles and corresponding opposite sides are A, B, C and a, b, c, respectively, the sine law is given by the formula...

## Derivation of Sum and Difference of Two Angles

The sum and difference of two angles can be derived from the figure shown below.

## Derivation of Formula for Sum of Years Digit Method (SYD)

The depreciation charge and the total depreciation at any time m using the sum-of-the-years-digit method is given by the following formulas:

**Depreciation Charge:**

**Total depreciation at any time m**

Where:

FC = first cost

SV = salvage value

n = economic life (in years)

m = any time before n (in years)

SYD = sum of years digit = 1 + 2 + ... + n = n(1 + n)/2

## Derivation of formula for volume of a frustum of pyramid/cone

**Frustum of a pyramid and frustum of a cone**

The formula for frustum of a pyramid or frustum of a cone is given by

Where:

h = perpendicular distance between A_{1} and A_{2} (h is called the altitude of the frustum)

A_{1} = area of the lower base

A_{2} = area of the upper base

Note that A_{1} and A_{2} are parallel to each other.

## Solution to Problem 341 | Torsion of thin-walled tube

**Problem 341**

Derive the torsion formula τ = Tρ / J for a solid circular section by assuming the section is composed of a series of concentric thin circular tubes. Assume that the shearing stress at any point is proportional to its radial distance.

## Derivation of the Double Angle Formulas

The Double Angle Formulas can be derived from Sum of Two Angles listed below:

$\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B$ → Equation (1)

$\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B$ → Equation (2)

$\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B}$ → Equation (3)

## Relationship Between Arithmetic Mean, Harmonic Mean, and Geometric Mean of Two Numbers

For two numbers *x* and *y*, let *x*, *a*, *y* be a sequence of three numbers. If *x*, *a*, *y* is an arithmetic progression then '*a*' is called *arithmetic mean*. If *x*, *a*, *y* is a geometric progression then '*a*' is called *geometric mean*. If *x*, *a*, *y* form a harmonic progression then '*a*' is called *harmonic mean*.

Let *AM* = arithmetic mean, *GM* = geometric mean, and *HM* = harmonic mean. The relationship between the three is given by the formula

Below is the derivation of this relationship.