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.
Similar Questions
- The adjacent sides of a parallelogram are $4a$ and $3a$. If the angle between them is $60^{\circ}$, then one of the diagonals of the parallelogram will be:
- Every interior angle of a regular octagon is $135^\circ$. Find the exterior angle of the octagon.
- PQ is a diameter of circle whose centre is O. If a point R lies on a circle and $\angle RPO$ is $42^\circ$, then find $\angle RQP$.
- ABC is an equilateral triangle and O is its circumcentre. If the side of triangle is 6 cm, then the $\angle BOC$ is:
- If every interior angle of a regular polygon is $144^\circ$, then the polygon has ______ sides.
- The ratio of the numbers of sides of two regular polygons is $1 : 2$. If each interior angle of the first polygon is $140^\circ$, then the measure of each interior angle of the second polygon is:
- Find the ratio of the measure of an angle of a regular pentagon to that of a regular octagon.
- What is the measure of each exterior angle of a regular octagon?
- The difference between the interior and exterior angles at a vertex of a regular polygon is $140^\circ$. The number of sides of the polygon is:
- The area of an isosceles right angle triangle is $81 \text{ cm}^2$ . Find the length of its hypotenuse.
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.
