Finding Regular Polygon Sides from Interior Angle
This solution details how to determine the number of sides of a regular polygon when given the measure of each interior angle.
Calculating the Exterior Angle
In any regular polygon, an interior angle and its adjacent exterior angle sum up to $180^{\circ}$.
Given interior angle $= 165^{\circ}$.
The formula for the exterior angle is:
Exterior Angle $= 180^{\circ} - \text{Interior Angle}$
Exterior Angle $= 180^{\circ} - 165^{\circ} = 15^{\circ}$.
Determining the Number of Sides
The sum of all exterior angles of any convex polygon is always $360^{\circ}$. For a regular polygon with $n$ sides, all exterior angles are equal. Therefore, the measure of one exterior angle is $\frac{360^{\circ}}{n}$.
Using the calculated exterior angle:
$15^{\circ} = \frac{360^{\circ}}{n}$
To find the number of sides ($n$), we rearrange the formula:
$n = \frac{360^{\circ}}{15^{\circ}}$
$n = 24$
The regular polygon has 24 sides.
Similar Questions
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Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
