On Superman and Pentagons

One of the most famous pentagons in existence is the one represented in Superman’s logo. This symbol is on Superman’s chest and cape, a pentagon with a yellow background and a red letter “S”. This pentagon in particular is unusual, it has two pairs of sides that are equal and one side that is not. This got me thinking about how to find the area of such an unusual shape, which I’m calling the Superman Pentagon.

Finding the area of a regular pentagon is not a formula that most people remember or learn about. A regular shape is any shape where the sides and angles are all equal, such as a square or equilateral triangle. When finding the area of any regular shape a triangle method is used. The shape is divided into a number of equal triangles, whose sides start at the corners of the shape and go to the center. The area of a triangle is used for one of these triangles and then multiplied by the number of triangles present. So, for a regular pentagon there are five of these triangles. The formula looks like this:

Area of a Triangle: 1/2 * (Base) * (Height)

Untitled

Area of a Regular Pentagon = 1/2 * (Base) * (Height) * 5

= 5/2 * (Base) * (Height)

However, the Superman pentagon is not a regular one. One method could be to divide the pentagon into triangles, but these triangles would not be the same size, and instead of a quick multiplication the sum of all of them would be needed.

This seems like too much work, so I wanted to investigate a new formula to determine the area for this particular polygon. The logo we will look at is one where the bottom half of the pentagon is an equilateral triangle, and a trapezoid sits above it. The formula for area would be the sum of the area of the triangle and the area of the trapezoid.

Area of a Triangle = 1/2 * B2*  H2

Area of a Trapezoid = 1/2 * (B1 +B2) * H1

 IMG_1875

Area of Superman’s Pentagon = 1/2*B2* H2 + 1/2 (B1+ B2) * H1

H = H1 + H2

Area of Superman’s Pentagon = 1/2(B2H+B1H1)

Note that the above is for a particular instance of the Superman logo. Over the course of the hero’s history there have been different variations and styles that may not follow this geometric pattern. It would be interesting to see if there were any ways to reduce or simplify the formula even further.

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