If four interior angles of a five-sided figure pentagon measure 100 degrees each, what will the fifth angle measure?

To understand why the fifth angle of this pentagon measures 140 degrees, we first need to know a fundamental rule of geometry: the sum of the interior angles of any polygon can be calculated with a simple formula. For a polygon with 'n' sides, the sum of its interior angles is (n-2) multiplied by 180 degrees. Since a pentagon has five sides, the sum of its interior angles is (5-2) x 180, which equals 540 degrees. Given that four of the angles are 100 degrees each, totaling 400 degrees, the fifth angle must be the remainder, which is 540 minus 400, or 140 degrees.

The name "pentagon" itself comes from the Greek words "pente" (meaning five) and "gonia" (meaning angle). While the figure in question is an irregular pentagon, with angles of different measures, a regular pentagon has five equal sides and five equal interior angles, each measuring 108 degrees. This five-sided shape is not just a geometric curiosity; it appears frequently in the world around us. You can see it in nature in the shape of flowers like morning glories and in the cross-section of an okra fruit.

Pentagons also hold cultural and architectural significance. The shape is famously used for the headquarters of the U.S. Department of Defense, known simply as the Pentagon, which is one of the world's largest office buildings. In various cultures, the pentagon has been used in intricate artistic patterns and holds symbolic meaning. From the home plate in baseball to the design of some forts, the five-sided figure is a testament to the blend of mathematics and practical design in our world.