Question

Find the measure of one interior angle and one exterior angle of each regular polygon. If necessary, round to the nearest degree. pentagon

Short answer

Interior angle is 108°, and exterior angle is 72°.

Step-by-step solution

Understand the Polygon

A regular pentagon is a five-sided polygon where all sides and angles are equal. This means that we need to calculate one interior and one exterior angle of this polygon.

Calculate the Sum of Interior Angles

The formula for the sum of the interior angles of a polygon is given by \((n-2) \times 180^\circ\), where \(n\) is the number of sides. For a pentagon, \(n = 5\). Thus, the sum is \((5-2) \times 180^\circ = 540^\circ\).

Find One Interior Angle

To find the measure of one interior angle in a regular pentagon, we divide the sum of interior angles by the number of sides. Therefore, each interior angle is \(\frac{540^\circ}{5} = 108^\circ\).

Calculate the Exterior Angle

The exterior angle of a regular polygon can be calculated by subtracting the measure of its interior angle from \(180^\circ\). Since we have found the interior angle to be \(108^\circ\), the exterior angle is \(180^\circ - 108^\circ = 72^\circ\).

Key concepts

Regular Polygon

A regular polygon is a unique type of polygon where all sides and all angles are equal. This characteristic makes calculations involving regular polygons much simpler. An easy way to recognize a regular polygon is by checking for symmetry. All the vertices and edges should be evenly spaced. Examples of regular polygons include equilateral triangles, squares, and regular pentagons.
Understanding regular polygons is essential when calculating angles as the symmetry ensures consistent internal and external angles. For instance, each angle in a regular pentagon is equal to others within the same shape, allowing straightforward mathematical approaches to find unknown values. This feature is particularly useful in the study of geometry and in applying polygon angle formulas.

Sum of Interior Angles

One important characteristic of polygons is the sum of their interior angles. The interior angles are the angles found on the inside of a polygon. The sum of these angles for any polygon can be calculated using the formula: \((n-2) \times 180^\circ\). Here, \(n\) is the number of sides of the polygon. This formula is derived from the fact that a polygon can be divided into \(n-2\) triangles.
The formula works for any polygon, whether regular or irregular, and provides a straightforward way to determine how angles are distributed within a shape. For a regular pentagon, which has 5 sides, the sum of the interior angles is \((5-2) \times 180^\circ = 540^\circ\). This means distributing these 540 degrees equally among the five angles of a regular pentagon guarantees each interior angle measures \(108^\circ\).
Knowing the sum of interior angles helps in understanding how a polygon's structure influences its overall geometry.

Polygon Angle Formulas

Polygon angle formulas are mathematical expressions that help us find the measure of interior and exterior angles. They facilitate understanding the properties of polygons and aid in geometric problem-solving.

• **Interior Angle Calculation**: To calculate one interior angle of a regular polygon with \(n\) sides, divide the sum of the interior angles by \(n\). The formula becomes: \(\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}\).
• **Exterior Angle Calculation**: The exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side. For regular polygons, each exterior angle can also be computed as \(180^\circ\) minus the interior angle. Equally, it can be calculated using the formula: \(\frac{360^\circ}{n}\), since the sum of the exterior angles for any polygon is always \(360^\circ\).

Using these formulas, we can efficiently solve problems like determining the angles in a regular pentagon, where we've found that each interior angle measures \(108^\circ\) and each exterior angle measures \(72^\circ\). Understanding these formulas enriches our comprehension of geometric properties and enhances our ability to work with various polygons.