Problem 88 - Prealgebra

Question

Geometry 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.

Step-by-Step Solution

Answer

Each angle of a regular pentagon measures 108°.

Step 1: Understand the Problem

We are given a regular pentagon, which means all its interior angles are equal. We need to determine the size of each angle.

Step 2: Recall the Formula for Interior Angles

The sum of the interior angles of a polygon with vertices is given by the formula \( (n-2) \times 180^{\circ} \), where \( n \) is the number of sides.

Step 3: Calculate the Total Sum of Interior Angles

Using the formula \((n-2) \times 180^{\circ} \), for a pentagon \(n=5\), so the sum is \( (5-2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ} \). The sum matches the given information.

Step 4: Determine the Measure of Each Angle

Since the pentagon is regular, divide the total sum of angles by the number of angles. Each angle is \( \frac{540^{\circ}}{5} = 108^{\circ} \).

Key Concepts

Understanding Interior Angles

Interior angles are the angles found on the inside of any polygon. Regardless of the number of sides, the interior angles in a polygon are consistently found using a polynomial formula. This is important when dealing with regular polygons, where all of the interior angles are equal. For any polygon with \(n\) sides, you can calculate the sum of the interior angles using the formula:

• \((n-2) \times 180^{\circ}\)

This formula arises from the idea that a polygon can be divided into triangles, each with a sum of angles equaling \(180^{\circ}\). By subtracting two from the number of sides, you calculate how many triangles can fit within the polygon.

Example in Practice

For example, a pentagon has five sides, so substituting into the formula gives:

• \((5-2) \times 180^{\circ} = 540^{\circ}\)

This tells us the total sum of the interior angles for any five-sided polygon.

Regular Pentagon

A regular pentagon is a special type of polygon where not only are all the sides equal, but all the interior angles are equal too. This makes solving for individual angles straightforward once you know the total sum of the interior angles.The pentagon is a five-sided figure, meaning the total sum of its interior angles can be calculated using the formula we discussed earlier:

• Total Sum = \(540^{\circ}\)

Since all angles in a regular pentagon are equal, you simply divide this total sum by the number of angles, which is also the number of sides in this case.

Calculating Specific Angles

A regular pentagon divides its total angle sum into five equal parts. Using the previous example, each interior angle will be:

• \(\frac{540^{\circ}}{5} = 108^{\circ}\)

Therefore, each angle in a regular pentagon measures \(108^{\circ}\).

Polygon Formula

The polygon formula is a vital tool in geometry that helps find the sum of the interior angles of any polygon. Using the formula:

• \((n-2) \times 180^{\circ}\)

We first deduct 2 from the number of sides, \(n\). This subtraction accounts for the two fewer sides needed to form triangles within the polygon. Each triangle contributes \(180^{\circ}\) to the total, multiplying by the number of triangles gives the sum of the polygon's interior angles.

Common Uses

- **Triangles**: With three sides, the formula gives a sum of \(180^{\circ}\), which is already well-known.- **Squares and Rectangles**: These have four sides, resulting in a sum of \(360^{\circ}\).- **Pentagons and Beyond**: For pentagons, like in our earlier example, you get \(540^{\circ}\).This formula is especially useful when dealing with regular polygons since it immediately allows the calculation of each angle when divided by the number of sides.