Calculating Polygon Sides from Angle Difference
Let the number of sides of the regular polygon be $n$. Let the interior angle be $I$ and the exterior angle be $E$. We know two key properties:
- The difference between the interior and exterior angle is given: $I - E = 140^\circ$.
- The sum of the interior and exterior angle at any vertex is $180^\circ$: $I + E = 180^\circ$.
Solving for Interior and Exterior Angles
We can solve these two equations simultaneously:
- Add the two equations: $(I - E) + (I + E) = 140^\circ + 180^\circ$ $2I = 320^\circ$ $I = \frac{320^\circ}{2} = 160^\circ$
- Substitute the value of $I$ into the second equation to find $E$: $160^\circ + E = 180^\circ$ $E = 180^\circ - 160^\circ = 20^\circ$
Determining the Number of Sides
The formula for the exterior angle of a regular polygon with $n$ sides is:
$E = \frac{360^\circ}{n}$We found that the exterior angle $E$ is $20^\circ$. Substitute this value:
$20^\circ = \frac{360^\circ}{n}$Rearrange the formula to solve for $n$:
$n = \frac{360^\circ}{20^\circ}$ $n = 18$Therefore, the regular polygon has 18 sides.
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:
- If the sum of the interior angles of a regular polygon is equal to four times the sum of its exterior angles, then what is the number of diagonals in the polygon?
- 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 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.
