The measure of each interior angle of a regular octagon is 135°.
Understanding Regular Octagons
An octagon is a fascinating polygon characterized by its eight sides, eight interior angles, and eight vertices. The term "regular" is crucial here, as it signifies that all sides of the octagon are equal in length, and all its interior angles are equal in measure. This uniformity simplifies the calculation of its angles.
Calculating the Interior Angle of a Regular Polygon
To determine the measure of each interior angle of any regular polygon, a specific formula can be applied. This formula is derived from the total sum of the interior angles of a polygon.
Formula for Interior Angles
The sum of the interior angles of any polygon with n sides is given by:
Sum of Interior Angles = $(n - 2) \times 180^\circ$
For a regular polygon, where all interior angles are equal, the measure of each individual interior angle is:
Each Interior Angle = $\frac{(n - 2) \times 180^\circ}{n}$
Where:
- n represents the number of sides (or angles) of the polygon.
Applying the Formula to an Octagon
For a regular octagon, the number of sides (n) is 8. Let's substitute this value into the formula:
-
Calculate the sum of interior angles:
Sum = $(8 - 2) \times 180^\circ$
Sum = $6 \times 180^\circ$
Sum = $1080^\circ$ -
Calculate each interior angle:
Each Angle = $\frac{1080^\circ}{8}$
Each Angle = $135^\circ$
Therefore, each interior angle of a regular octagon measures 135°.
Key Properties of a Regular Octagon
| Property | Value |
|---|---|
| Number of Sides (n) | 8 |
| Number of Vertices | 8 |
| Number of Interior Angles | 8 |
| Sum of Interior Angles | $1080^\circ$ |
| Each Interior Angle | $135^\circ$ |
| Each Exterior Angle | $45^\circ$ (since $180^\circ - 135^\circ$) |
Practical Insights and Examples
Regular octagons are common in everyday life and design due to their balanced and symmetrical form.
- Stop Signs: Perhaps the most universally recognized example of a regular octagon is the standard stop sign. Its octagonal shape is not just for aesthetics; it's a critical safety feature, ensuring it's recognizable from any angle, even if partially obscured or viewed from the back.
- Architecture: Octagonal designs appear in various architectural elements, from windows and towers to gazebos and flooring patterns, adding a unique aesthetic appeal.
- Gaming: Certain game boards or tiles may utilize octagonal shapes to create interesting tessellations or playing areas.
The consistent 135° angle for each interior corner is what gives the regular octagon its distinctive and balanced appearance.
Summary of Angle Measure
To reiterate, a regular octagon is a polygon with eight equal sides and eight equal interior angles. Each of these interior angles precisely measures 135°.
