Question
If an exterior angle of a regular polygon is 40°, then find the number of diagonals of the polygon.
The correct answer is
27
Calculating Polygon Diagonals from Exterior Angle
The problem asks us to find the number of diagonals of a regular polygon given that one of its exterior angles measures 40°.
Step 1: Determine the Number of Sides
For any regular polygon, the sum of the exterior angles is 360°. If the polygon has n sides, each exterior angle measures $ \frac{360°}{n} $. We are given that the exterior angle is 40°.
Set up the equation:
$ \frac{360°}{n} = 40° $Solve for n:
$ n = \frac{360°}{40°} $ $ n = 9 $Therefore, the regular polygon has 9 sides (it is a nonagon).
Step 2: Calculate the Number of Diagonals
The formula for the number of diagonals (D) in a polygon with n sides is:
$ D = \frac{n(n-3)}{2} $Substitute the value of n = 9 into the formula:
$ D = \frac{9(9-3)}{2} $ $ D = \frac{9(6)}{2} $ $ D = \frac{54}{2} $ $ D = 27 $The polygon has 27 diagonals.
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Important Questions from Geometry (Notes)
- A 6 cm long chord of a circle is at a distance of 4 cm from the centre of the circle. Find the distance of 8 cm long chord of the same circle from the centre.
- In a triangle PQR, if $\angle P + \angle R = 150^\circ$ and $\angle P + 3\angle Q = 170^\circ$, then $\angle P$ is equal to :
- PQR is a triangle. The bisectors of the internal angle $\angle Q$ and external angle $\angle R$ intersect at M. If $\angle QMR = 40^\circ$, then $\angle P$ is :
- O and C are respectively the orthocentre and the circumcentre of an acute angled triangle $\triangle PQR$. $\angle QCR = 138^\circ$. A perpendicular PM is dropped from P on side QR. If $\angle PQR = 52^\circ$, what is the degree measure of $\angle RPM$?
- A $2\text{ m}$ long ladder is to reach a wall of height $1.75\text{ m}$. The largest possible horizontal distance of the ladder from the wall could be
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