Problem 84 - Prealgebra

Question

This Google Earth image shows the Pentagon. The interior angles of a regular pentagon are all the same and sum to \(540^{\circ} .\) Find the size of each angle. CAN'T COPY THE IMAGE

Step-by-Step Solution

Answer

Each angle of a regular pentagon is \(108^{\circ}\).

Step 1: Understanding a Regular Pentagon

A regular pentagon is a five-sided polygon with all sides of equal length and all interior angles of equal measure. Given that the sum of the interior angles of any pentagon is \(540^{\circ}\), we can use this information to find each individual angle.

Step 2: Use the Interior Angle Sum Formula

The formula for the sum of the interior angles of a polygon is \((n - 2) \times 180^{\circ}\), where \(n\) is the number of sides. For a pentagon, \(n = 5\), so the sum is \((5 - 2) \times 180^{\circ} = 540^{\circ}\). This verifies that the given information is correct.

Step 3: Calculate Each Interior Angle

Since the pentagon is regular, each interior angle is the same. To find the size of each angle, divide the total sum of the interior angles by the number of angles: \(\frac{540^{\circ}}{5} = 108^{\circ}\).

Key Concepts

Interior Angles

The interior angles of a polygon are those angles found on the inside of the shape, formed between two sides. For a regular pentagon, these angles are all equal in measure. Given that the sum of the interior angles of a regular pentagon is known to be \(540^{\circ}\), it simplifies the task of finding each angle.

• "Interior" refers to the inside of the shape.
• In a regular polygon, all interior angles are of equal measure.

To find the measure of one interior angle in a regular pentagon, you can divide the total sum of the angles by the number of angles in the polygon. This equality of angles in regular polygons helps in understanding their geometrical properties easily.

Polygon

A polygon is a flat, two-dimensional shape with straight sides. It can have any number of sides, but they must be at least three to form a closed shape. Polygons are usually named according to the number of sides they have: triangle (3 sides), quadrilateral (4 sides), pentagon (5 sides), and so on. Regular polygons have sides of equal length and angles of equal measure.

• Regular polygons have all sides and angles equal.
• The name of a polygon indicates the number of sides it has.

Understanding that a pentagon is a type of polygon with five sides, and that a regular pentagon further has all sides and interior angles equal, provides the foundation for calculating angles within it.

Angle Sum Formula

The angle sum formula is a crucial tool for determining the sum of the interior angles of a polygon. This formula is given by \((n - 2) \times 180^{\circ}\), where \(n\) represents the number of sides in the polygon.

• For a triangle (3 sides), the sum is \(180^{\circ}\).
• For a quadrilateral (4 sides), it is \(360^{\circ}\).
• For a pentagon (5 sides), the sum is \(540^{\circ}\).

This formula provides an easy method to verify the total interior angles of any polygon. Applying it to a pentagon shows that the total is indeed \(540^{\circ}\), which helps checking calculated values and supports understanding of geometric concepts.

Prealgebra

Prealgebra involves the basic mathematical skills necessary to solve problems involving integers, fractions, decimals, and simple geometric shapes. It bridges the gap between basic arithmetic and more advanced algebra concepts. When tackling geometry problems like those involving the angles of a polygon, prealgebraic skills are crucial.

• Understanding geometric shapes and their properties.
• Applying formulas to solve problems.
• Working with fractions and division to find individual angles.

Engaging with regular pentagons and their properties, such as calculating the size of each interior angle, is an excellent way to apply prealgebra concepts in real-world situations. The division of the total angle sum by the number of sides tests these basic mathematical skills effectively.