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

A five-sided figure, known as a pentagon, always has interior angles that sum to a total of 540 degrees. This holds true whether the pentagon is regular, with all equal sides and angles, or irregular, where the sides and angles can vary. If you know the measurements of four of these angles, you can easily find the fifth by subtracting the sum of the known angles from 540 degrees. For instance, if four angles each measure 100 degrees, their sum is 400 degrees. Subtracting 400 from 540 leaves 140 degrees for the final angle.

This 540-degree rule isn't arbitrary; it stems from a fundamental principle of polygon geometry. The sum of the interior angles of any polygon can be found using the formula (n-2) * 180 degrees, where 'n' represents the number of sides. For a pentagon, with five sides, the calculation becomes (5-2) * 180, which simplifies to 3 * 180, resulting in 540 degrees. This formula works because any polygon can be divided into a certain number of triangles by drawing diagonals from a single vertex, and each triangle's interior angles sum to 180 degrees. A pentagon can be divided into three such triangles.

Pentagons are more than just mathematical exercises; they appear in many aspects of the world around us. From the iconic Pentagon building in the United States, which serves as the headquarters for the Department of Defense, to the familiar shape of home plate on a baseball field, and even the patterns on a soccer ball, this five-sided shape is quite common. Understanding its properties, like the sum of its interior angles, helps us appreciate the geometry inherent in both natural and man-made designs.