Calculating Regular Pentagon Angle
The formula for the measure of an interior angle of a regular n-sided polygon is:
$ \text{Angle} = \frac{(n-2) \times 180^\circ}{n} $For a regular pentagon, n = 5.
Substituting n = 5 into the formula:
$ \text{Angle}_{\text{pentagon}} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ $Calculating Regular Octagon Angle
For a regular octagon, n = 8.
Substituting n = 8 into the formula:
$ \text{Angle}_{\text{octagon}} = \frac{(8-2) \times 180^\circ}{8} = \frac{6 \times 180^\circ}{8} = \frac{1080^\circ}{8} = 135^\circ $Determining the Angle Ratio
The question asks for the ratio of the measure of an angle of a regular pentagon to that of a regular octagon.
Ratio = $ \frac{\text{Angle}_{\text{pentagon}}}{\text{Angle}_{\text{octagon}}} = \frac{108^\circ}{135^\circ} $
To simplify the ratio, find the greatest common divisor (GCD) of 108 and 135, which is 27.
Divide both numbers by the GCD:
$ \frac{108 \div 27}{135 \div 27} = \frac{4}{5} $The ratio is 4:5.
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?
- 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 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 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
