Calculating Polygon Sides from Angle Difference
This problem involves finding the number of sides ($n$) of a regular polygon based on the difference between its interior and exterior angles at a vertex.
Understanding Angle Formulas
For a regular polygon with $n$ sides:
- The exterior angle ($E$) is given by the formula: $E = \frac{360^\circ}{n}$
- The interior angle ($I$) is related to the exterior angle by: $I = 180^\circ - E$
Solving for the Number of Sides
We are given that the difference between the interior and exterior angles is $160^\circ$. So:
$I - E = 160^\circ$Substitute the expression for $I$ ($I = 180^\circ - E$) into the equation:
$(180^\circ - E) - E = 160^\circ$ $180^\circ - 2E = 160^\circ$Now, solve for the exterior angle $E$:
$2E = 180^\circ - 160^\circ$ $2E = 20^\circ$ $E = \frac{20^\circ}{2}$ $E = 10^\circ$Finally, use the formula for the exterior angle to find the number of sides ($n$):
$E = \frac{360^\circ}{n}$ $10^\circ = \frac{360^\circ}{n}$Rearrange the formula to solve for $n$:
$n = \frac{360^\circ}{10^\circ}$ $n = 36$Therefore, the regular polygon has 36 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 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:
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.
