What is the magnitude (in radian) of the interior angle of a regular pentagon?

Question

Prepp Team · Accepted Answer

Understanding the interior angles of regular polygons is a fundamental concept in geometry. A regular pentagon is a polygon with five equal sides and five equal interior angles.

Calculating the Sum of Interior Angles of a Pentagon

The formula for the sum of the interior angles of any polygon with $n$ sides is given by:

$\text{Sum of interior angles} = (n-2) \times 180^\circ$

For a pentagon, the number of sides $n=5$. Substituting this value into the formula:

$\text{Sum} = (5-2) \times 180^\circ$

$\text{Sum} = 3 \times 180^\circ$

$\text{Sum} = 540^\circ$

So, the total sum of the interior angles of a pentagon is $540^\circ$.

Finding the Magnitude of One Interior Angle

Since a regular pentagon has five equal interior angles, we can find the magnitude of a single interior angle by dividing the sum by the number of sides (or angles):

$\text{Magnitude of one interior angle} = \frac{\text{Sum of interior angles}}{\text{Number of sides}}$

$\text{Magnitude} = \frac{540^\circ}{5}$

$\text{Magnitude} = 108^\circ$

Thus, each interior angle of a regular pentagon is $108^\circ$.

Converting Degrees to Radians

The question asks for the angle in radians. To convert an angle from degrees to radians, we use the conversion factor that $180^\circ = \pi$ radians. The formula for conversion is:

$\text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180^\circ}$

Now, we convert the interior angle of $108^\circ$ to radians:

$\text{Angle in radians} = 108^\circ \times \frac{\pi}{180^\circ}$

$\text{Angle in radians} = \frac{108\pi}{180}$

Simplifying the Radian Measure

We need to simplify the fraction $\frac{108}{180}$. We can find common factors to divide both the numerator and the denominator. Both are divisible by 2, 4, 6, 9, 12, 18, 36. The greatest common divisor is 36.

Divide the numerator by 36:

$108 \div 36 = 3$

Divide the denominator by 36:

$180 \div 36 = 5$

So, the simplified fraction is $\frac{3}{5}$. Therefore, the interior angle in radians is $\frac{3\pi}{5}$.

The magnitude of the interior angle of a regular pentagon is $\frac{3\pi}{5}$ radians.

Polygon	Number of Sides (n)	Sum of Interior Angles (Degrees)	Each Interior Angle (Degrees)	Each Interior Angle (Radians)
Equilateral Triangle	3	(3-2) * 180° = 180°	180° / 3 = 60°	60 * π/180 = π/3
Square	4	(4-2) * 180° = 360°	360° / 4 = 90°	90 * π/180 = π/2
Regular Pentagon	5	(5-2) * 180° = 540°	540° / 5 = 108°	108 * π/180 = 3π/5
Regular Hexagon	6	(6-2) * 180° = 720°	720° / 6 = 120°	120 * π/180 = 2π/3

Revision Table: Polygon Angle Formulas

Concept	Formula
Sum of Interior Angles (n-sided polygon)	$(n-2) \times 180^\circ$ or $(n-2) \times \pi$ radians
Each Interior Angle (n-sided regular polygon)	$\frac{(n-2) \times 180^\circ}{n}$ or $\frac{(n-2) \times \pi}{n}$ radians
Sum of Exterior Angles (any polygon)	$360^\circ$ or $2\pi$ radians
Each Exterior Angle (n-sided regular polygon)	$\frac{360^\circ}{n}$ or $\frac{2\pi}{n}$ radians
Relationship between Interior and Exterior Angle (at a vertex)	Interior angle + Exterior angle = $180^\circ$ or $\pi$ radians

Additional Information: Regular Polygons and Angles

A regular polygon is a polygon that is both equilateral (all sides are equal) and equiangular (all interior angles are equal).

The concept of interior angles is closely related to exterior angles. An exterior angle of a convex polygon is the angle formed by one side and the extension of an adjacent side.

For any convex polygon, the sum of the exterior angles is always $360^\circ$ (or $2\pi$ radians).
For a regular n-sided polygon, each exterior angle is $\frac{360^\circ}{n}$ or $\frac{2\pi}{n}$ radians.
You can also find the interior angle using the exterior angle: Interior angle = $180^\circ$ - Exterior angle (in degrees) or Interior angle = $\pi$ - Exterior angle (in radians).

Let's check this for the regular pentagon using radians:

Each exterior angle of a regular pentagon is $\frac{2\pi}{5}$ radians.
Interior angle = $\pi - \frac{2\pi}{5} = \frac{5\pi}{5} - \frac{2\pi}{5} = \frac{3\pi}{5}$ radians. This matches our previous calculation.

Understanding both interior and exterior angles provides a complete picture of the angles within and around a polygon.

Prepp Team · Answer

$\frac{\pi}{5}$

Prepp Team · Answer

$\frac{2\pi}{5}$

Prepp Team · Answer

$\frac{4\pi}{5}$

Calculating the Sum of Interior Angles of a Pentagon

Finding the Magnitude of One Interior Angle

Converting Degrees to Radians

Simplifying the Radian Measure

Revision Table: Polygon Angle Formulas

Additional Information: Regular Polygons and Angles

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Concept	Formula
Sum of Interior Angles (n-sided polygon)	\((n-2) \times 180^\circ\) or \((n-2) \times \pi\) radians
Each Interior Angle (n-sided regular polygon)	\(\frac{(n-2) \times 180^\circ}{n}\) or \(\frac{(n-2) \times \pi}{n}\) radians
Sum of Exterior Angles (any polygon)	\(360^\circ\) or \(2\pi\) radians
Each Exterior Angle (n-sided regular polygon)	\(\frac{360^\circ}{n}\) or \(\frac{2\pi}{n}\) radians
Relationship between Interior and Exterior Angle (at a vertex)	Interior angle + Exterior angle = \(180^\circ\) or \(\pi\) radians