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
- 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.
- The area of an isosceles right angle triangle is $81 \text{ cm}^2$ . Find the length of its hypotenuse.
- In a polygon, the sum of the interior angles is triple the sum of the exterior angles. The number of sides is:
- What is the measure of each exterior angle of a regular octagon?
- The ratio of an interior angle to the exterior angle of a regular polygon is $4:1$. The number of sides of the polygon is:
- If the interior angles of a pentagon are in the ratio 1 : 3 : 5 : 7 : 11, then the measure of the smallest interior angle is:
- If the side of an equilateral triangle is 2 cm, then find the area and the altitude of the triangle.
- Two sides of a triangle are of lengths 4 cm and 10 cm. If the length of the third side is $a$ cm, then:
- If the difference between the interior and exterior angles of a polygon is $36^\circ$, then find the number of sides in the polygon.
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
