On your paper draw a line segment which is 10 inches long. Now you draw all the points on the paper which are exactly 5 inches from the line, and you will get a closed figure. What is the area of this closed figure?

This geometric puzzle creates a shape familiar to many, often called a capsule or a stadium. The common mistake is to only visualize two parallel lines, one above and one below the original segment. This would indeed form a rectangle. However, the prompt asks for all points exactly 5 inches from the line *segment*, which includes its two endpoints. The set of points exactly 5 inches from an endpoint forms a semicircle, creating the rounded ends of the capsule shape.

To find the area, we can deconstruct this figure into two simple components: a central rectangle and two semicircles. The rectangle's length is the same as the original segment, 10 inches. Its width is 10 inches (5 inches "above" the line and 5 inches "below"), giving it an area of 100 square inches.

The two semicircles, one at