A Proof that the Vertex Angle Sum of a Pentagram is 180 degrees

The pentagram is a five-pointed star. It was used by the ancient Greeks as a symbol of faith.  In this post, we exhibit the mathematics of pentagrams — we show that the sum of the angle measures of its vertices equals 180°.

pentagram math

For regular pentagrams, the proof is simple. By the inscribed angle theorem, the measure of an inscribed angle is half the measure of a central angle that intercepts the same arc. The central angles of a regular pentagram as shown above intercept the entire circle. Therefore, its angle measures add up to 360°.

The vertex angles, on the other hand, are inscribed angles as shown in the second image above. Notice that if we add them up, they also intercept the entire circle (Can you see why?).  In effect,  their angle sum is half of 360°,  which equals 180°. Continue reading

GeoGebra launches official blog and more

To all GeoGebra fans out there, GeoGebra has recently launched its official blog. The blog will contain news, tricks, tips, and everything about GeoGebra. According to the administrator, guest posts are expected from GeoGebra bloggers all over the world.

I took  a one month leave from GeoGebra Applet Central. I thought I needed a vacation, but I’m resuming GeoGebra blogging next week.  I am also planning to revise the remaining GeoGebra tutorials that have not been updated to GeoGebra 4.0.

***

For a while now, I have been reading Ask a Mathematician/Ask A Physicist blog. It’s a great blog that discusses questions in mathematics and physics. Although, the blog is not really intended for high school, some senior high school and Math and Physics geeks might appreciate it. Continue reading

7 Extraordinary Mathematicians You Should Know About

There are numerous mathematicians who have made significant contributions in the field of mathematics. We cannot argue the mathematical greatness of Euclid, Newton, Gauss, Euler, and others who have set the foundation to the many branches of mathematics. In this post, we learn about 7 extraordinary mathematicians who are quite less known — less known in the sense that they are probably familiar to those who study mathematics and related fields.

1. Evariste Galois (1811-1832, France)
Evariste Galois was probably the most unfortunate mathematician who ever lived. He lived during the political turmoil in France. He failed the entrance examinations at Ecole Polytechnique twice because he could not explain his answers, was jailed for six months, and died in a duel at the age of 21.

Evariste Galois

Galois was  ahead of his time. In his teens, he was able to determine necessary and sufficient conditions for algebraic solutions of polynomials to exist. He barely attended college, but most of his contemporaries could not understand his work. He submitted research papers that were either lost or “incomprehensible.”  It was only 14 years after his death that the mathematics community was able to recognize the value of his work.

Despite his short life and his numerous misfortunes, his works gave a firm foundation to group theory. Continue reading

Visualizing the Difference of Two Squares

Geometric objects are powerful representations that can be used to visualize algebraic properties of mathematical objects. Proofs without words are examples of such visual representations. In this post, we relate the difference of two squares to the areas of squares and rectangles.

The difference of two squares  states that for all numbers a and ba^2 - b^2 = (a + b)(a-b). The visual representation below, however, only covers the condition that a^2 - b^2 \geq 0 (Why?).

difference of two squares

To proceed with the visual proof, we create a square with side length a as shown in (1). Then, we cut a square with side length b from its corner as shown in (2). Since the area of the larger square is a^2 and the area of the smaller square is b^2, the area of the remaining figure is a^2 - b^2. Continue reading

Related Posts Plugin for WordPress, Blogger...