Polygon Diagonals from Angle Sums
The problem involves finding the number of diagonals in a regular polygon based on the relationship between the sum of its interior angles and the sum of its exterior angles.
Finding the Number of Sides (n)
We use the standard formulas for the sum of interior and exterior angles of a polygon with n sides:
- Sum of interior angles = $\(n-2) \times 180^\circ$
- Sum of exterior angles = $\360^\circ$
The problem states that the sum of the interior angles is six times the sum of the exterior angles:
$\(n-2) \times 180^\circ = 6 \times 360^\circ$
To find n, we solve this equation:
- Divide both sides by $\180^\circ$: $\n-2 = 6 \times \frac{360^\circ}{180^\circ}$ $\n-2 = 6 \times 2$ $\n-2 = 12$
- Add 2 to both sides: $\n = 12 + 2$ $\n = 14$
The polygon has 14 sides.
Calculating the Number of Diagonals
The formula for the number of diagonals in a polygon with n sides is:
Number of diagonals = $\\frac{n(n-3)}{2}$
Substitute the value $\n=14$ into the formula:
Number of diagonals = $\\frac{14(14-3)}{2}$
Number of diagonals = $\\frac{14 \times 11}{2}$
Number of diagonals = $\\frac{154}{2}$
Number of diagonals = $\77$
Therefore, the polygon has 77 diagonals.
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.
