Regular Pentagon Center Angle Calculation
A regular pentagon is a polygon with 5 equal sides and 5 equal interior angles. An inscribed circle fits perfectly inside, touching the midpoint of each side. The center of this circle is the same as the center of the pentagon.
To find the angle subtended by one side at the center, we consider the lines connecting the center to the two vertices forming that side. These lines, along with the side itself, form an isosceles triangle.
Determining the Central Angle
The sum of angles around any point (like the center of the pentagon) is always $360^\circ$. A regular pentagon can be divided into 5 identical isosceles triangles, with their vertices meeting at the center.
The angle subtended by each side at the center is equal because the pentagon is regular.
Calculation Steps:
- Total angle around the center: $360^\circ$
- Number of sides in a regular pentagon: 5
- Angle subtended by one side = $\frac{\text{Total angle}}{\text{Number of sides}}$
- Angle = $\frac{360^\circ}{5}$
- Angle = $72^\circ$
Therefore, the angle subtended by one side of the regular pentagon at the center of the inscribed circle is $72^\circ$.
Similar Questions
- The number of triangles formed by diagonals from one vertex of an octagon is:
- The sum of all interior angles of a 20-sided polygon is:
- If the measure of an exterior angle of a regular polygon is $20^\circ$, determine the total number of its sides.
- What is the minimum distance between centers of two circles having radii 5 cm and 3 cm such that exactly three common tangents exist?
- In a circle, angle subtended by chord AB at the center is $80^\circ$. What is the measure of angle $\angle ACB$, where C lies on the circle?
- In a circle with center O, two chords, AB and CD, cross each other at right angles. if the distance from the center O to chord AB is 3 cm, while the distance from O to chord CD is 4 cm. What is the radius of the circle?
- If the angle subtended by a chord at the center is $150^\circ$, what is the angle subtended at a point on the circle in the same segment?
- A triangle ABC has a circle inscribed within it with sides a, b, c. The area of the triangle is A. What is the radius of the inscribed circle?
- In a triangle ABC, if the angle bisectors of $\angle B$ and $\angle C$ intersect at point I, and $\angle BIC = 110^\circ$, what is the measure of $\angle A$?
- In triangle PQR, ST is parallel to QR. If PS=5cm, SQ=10cm, and PT=6cm, find TR.
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.
