Sum of Circumference af all the Circles

A circle of radius 1 inch is inscribed in an equilateral triangle. A smaller circle is inscribed at each vertex, tangent to the circle and two sides of the triangle. The process is continued with progressively smaller circles. What is the sum of the circumference of all the circles.



Files for Download are Now Available to Non-logged-in Users

MATHalino's philosophy, I may say, is that our contents must be accessible to all. Whether you are logged in or not, you can still access all our contents without any constraint. We already done this last year to our Courses and Exams, and this year we extend it to downloads.



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 $r$

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


Factor trinomials mentally! Tips and tricks

Here are some tips and tricks in factoring the trinomial $ax^{2}+bx+c$ mentally. Once you master the techniques in this blog, you can simplify expressions and solve equations that require factoring with "lightning" speed, and impress your friends.

A triangular shaped channel is to be designed to carry 700 L/s on a slope of 0.0001. Determine what vertex angle and depth of water over the vertex will be necessary to give a section with minimum perimeter, assuming the channel is made of timber, n = 0.012. Use Manning’s formula.

A.   θ = 45°, h = 1.425 m C.   θ = 45°, h = 2.125 m
B.   θ = 90°, h = 2.215 m D.   θ = 90°, h = 1.215 m