Calculating Polygon Diagonals Based on Angle Sums
This problem requires us to find the number of diagonals in a regular polygon given a relationship between its interior and exterior angle sums.
Step 1: Relate Interior and Exterior Angle Sums
Let $n$ be the number of sides of the regular polygon.
- The sum of the interior angles of an $n$-sided polygon is given by the formula: $ \text{Sum}_{\text{interior}} = (n-2) \times 180^\circ $.
- The sum of the exterior angles of any convex polygon is constant: $ \text{Sum}_{\text{exterior}} = 360^\circ $.
The problem states that the sum of the interior angles is four times the sum of its exterior angles:
$ (n-2) \times 180^\circ = 4 \times 360^\circ $Step 2: Solve for the Number of Sides (n)
Simplify the equation:
$ (n-2) \times 180 = 1440 $Divide both sides by 180:
$ n-2 = \frac{1440}{180} $ $ n-2 = 8 $Add 2 to both sides:
$ n = 8 + 2 $ $ n = 10 $The polygon has 10 sides.
Step 3: Calculate the Number of Diagonals
The formula for the number of diagonals ($D$) in an $n$-sided polygon is:
$ D = \frac{n(n-3)}{2} $Substitute $n=10$ into the formula:
$ D = \frac{10(10-3)}{2} $ $ D = \frac{10 \times 7}{2} $ $ D = \frac{70}{2} $ $ D = 35 $Therefore, the polygon has 35 diagonals.
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Important Questions from Geometry
If the angles of a triangle are in the ratio of 2:3:4, then the difference of the measure of greatest angle and smallest angle is
In ΔABC, ∠A = 90°, AD ┴ BC and AD = BD = 2 cm. The length of CD is
How many lines of symmetry does a rectangle have?
The number of diagonals in each face a cube is
Which of the following is/are the geometric figures with the line of symmetry?
I. Rectangle
II. Isosceles triangle
