What is the interior angle of a regular octagon of side length 2 cm?
Calculating the Interior Angle of a Regular Octagon
The question asks for the interior angle of a regular octagon. A regular polygon is a polygon where all sides are equal in length and all interior angles are equal in measure. An octagon is a polygon with 8 sides. The side length given (2 cm) is not needed to calculate the interior angle; the number of sides is sufficient.
Formula for Interior Angle
The formula to find the measure of each interior angle of a regular polygon with \(n\) sides is given by:
\(\text{Interior Angle} = \dfrac{(n-2) \times 180^\circ}{n}\)
Alternatively, if the answer is required in radians, the formula is:
\(\text{Interior Angle} = \dfrac{(n-2)\pi}{n}\) radians
Applying the Formula to a Regular Octagon
For a regular octagon, the number of sides, \(n\), is 8.
Using the formula in radians:
- Substitute \(n=8\) into the formula:
- \(\text{Interior Angle} = \dfrac{(8-2)\pi}{8}\)
- Simplify the expression:
- \(\text{Interior Angle} = \dfrac{6\pi}{8}\)
- Reduce the fraction:
- \(\text{Interior Angle} = \dfrac{3\pi}{4}\) radians
To express this in degrees for better understanding, we can convert radians to degrees (since \(\pi\) radians = \(180^\circ\)):
\(\text{Interior Angle} = \dfrac{3}{4} \times 180^\circ\)
\(\text{Interior Angle} = 3 \times 45^\circ\)
\(\text{Interior Angle} = 135^\circ\)
The options provided are in radians, and our calculated angle is \(\dfrac{3\pi}{4}\) radians.
Summary of Calculation
| Polygon Type | Number of Sides (n) | Interior Angle Formula (Radians) | Calculation | Interior Angle (Radians) |
|---|---|---|---|---|
| Regular Octagon | 8 | \(\dfrac{(n-2)\pi}{n}\) | \(\dfrac{(8-2)\pi}{8} = \dfrac{6\pi}{8}\) | \(\dfrac{3\pi}{4}\) |
Therefore, the interior angle of a regular octagon is \(\dfrac{3\pi}{4}\) radians.
Revision Table: Regular Polygons and Angles
| Polygon | Number of Sides (n) | Sum of Interior Angles (\((n-2) \times 180^\circ\)) | Each Interior Angle (Degrees) | Each Interior Angle (Radians) |
|---|---|---|---|---|
| Triangle | 3 | \(1 \times 180^\circ = 180^\circ\) | \(180^\circ / 3 = 60^\circ\) | \(\pi / 3\) |
| Square | 4 | \(2 \times 180^\circ = 360^\circ\) | \(360^\circ / 4 = 90^\circ\) | \(\pi / 2\) |
| Pentagon | 5 | \(3 \times 180^\circ = 540^\circ\) | \(540^\circ / 5 = 108^\circ\) | \(3\pi / 5\) |
| Hexagon | 6 | \(4 \times 180^\circ = 720^\circ\) | \(720^\circ / 6 = 120^\circ\) | \(2\pi / 3\) |
| Octagon | 8 | \(6 \times 180^\circ = 1080^\circ\) | \(1080^\circ / 8 = 135^\circ\) | \(3\pi / 4\) |
| Decagon | 10 | \(8 \times 180^\circ = 1440^\circ\) | \(1440^\circ / 10 = 144^\circ\) | \(4\pi / 5\) |
Additional Information: Exterior Angles
For any convex polygon, the sum of the exterior angles is always \(360^\circ\) or \(2\pi\) radians.
For a regular polygon with \(n\) sides, each exterior angle is equal to:
\(\text{Each Exterior Angle} = \dfrac{360^\circ}{n}\) or \(\dfrac{2\pi}{n}\) radians
Also, the interior angle and its adjacent exterior angle at any vertex sum up to \(180^\circ\) or \(\pi\) radians. This provides an alternative way to calculate the interior angle:
\(\text{Interior Angle} = 180^\circ - \text{Exterior Angle}\)
or
\(\text{Interior Angle} = \pi - \text{Exterior Angle}\)
For a regular octagon (n=8):
- Each Exterior Angle = \(\dfrac{2\pi}{8} = \dfrac{\pi}{4}\) radians
- Interior Angle = \(\pi - \dfrac{\pi}{4} = \dfrac{4\pi - \pi}{4} = \dfrac{3\pi}{4}\) radians
This confirms the result obtained using the interior angle formula directly.
Similar Questions
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Important Questions from Polygons
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