## 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.

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

$R = \dfrac{abc}{4A_t}$

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

The radius of incircle is given by the formula

$r = \dfrac{A_t}{s}$

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

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

$a^2 = b^2 + c^2 - 2bc\cos A$

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

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

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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...

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

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

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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:**

$d_m = (FC - SV) \dfrac{n - m + 1}{SYD}$

**Total depreciation at any time m**

$D_m = (FC - SV) \dfrac{m(2n - m + 1)}{2 \times SYD}$

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

$V = \dfrac{h}{3} \left[ \, A_1 + A_2 + \sqrt{A_1A_2} \, \right]$

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.

**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)

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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

$AM \times HM = GM^2$

Below is the derivation of this relationship.