Derivation of Formula

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.
 

The radius of incircle is given by the formula
 

where A_t = area of the triangle and s = semi-perimeter.
 

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

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...
 

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

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
 

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 A₁ and A₂ (h is called the altitude of the frustum)
A₁ = area of the lower base
A₂ = area of the upper base
Note that A₁ and A₂ are parallel to each other.
 

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.
 

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)
 

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.