Problem 20 Find the area of a regular decag... [FREE SOLUTION] (2024)

Short Answer

geometry

Geometry is a branch of mathematics that studies the sizes, shapes, and properties of figures and spaces. It helps in understanding the forms around us and the concepts of space and distance. Regular polygons, like the regular decagon in this exercise, are a fundamental part of geometry. In a regular polygon, all sides and angles are equal. Understanding geometry helps in calculating areas, volumes, and other properties of various shapes.

regular decagon

A regular decagon is a polygon with ten equal sides and ten equal angles. Each internal angle in a regular decagon is \(144\text{ degrees}\). Regular decagons can be found in various geometrical problems and have applications in design and architecture. Because all sides and angles are equal, it simplifies calculations such as finding the area or perimeter. In our exercise, the given decagon has all sides of equal length and all angles equal.

area calculation

To calculate the area of a regular polygon, you can use the formula: \[ \text{Area} = \frac{1}{2} \times \text{Apothem} \times \text{Perimeter} \] The area gives the amount of space inside the boundary of a polygon. It is measured in square units. In this exercise, we used the given apothem (5.7 cm) and perimeter (37 cm) of the decagon. By substituting these values, we calculated the area step-by-step for clarity. Always remember to double-check your units and values while performing these calculations.

apothem

The apothem of a polygon is the line from the center to the midpoint of one of its sides. In a regular polygon, the apothem is always perpendicular to the side. It can also be defined as the radius of the inscribed circle of the polygon. In this exercise, the given apothem is 5.7 cm. The apothem is an essential part of calculating the area as it reflects the central spacing and helps in deriving the formula for the area.

perimeter

The perimeter of a polygon is the total length around the polygon. It is found by adding the lengths of all sides. For regular polygons, you can multiply the length of one side by the number of sides. In this exercise, the perimeter of the decagon is 37 cm. The perimeter, together with the apothem, is used in the area formula. Being precise with measuring the perimeter aids in accurately determining the area.