1.Q.
Find the number of diagonals in an octagon?

Where n is
Number Of Sides

Here n = 8

Diagonals=
[8(8-3)5]/2 = 20

2.Q. Find the number of sides of a polygon whose
each exterior angle is 45

^{0}.^{ }
Ans: Measure of
Each Exterior Angle of a Polygon = 360/n

Each Exterior
Angle = 45

45 = 360/n

Number of
Sides = 360/45 =8

So Number of
Sides = 8

3. Q. The sum of the interior angles of a regular polygon is 3 times
the sum of its exterior angles. Determine the number of sides of the polygon.

Ans: sum of
the interior angles of a regular polygon is 3 times the sum of its exterior
angles.

We know that
in a regular polygon sum of all the exterior angles = 360°

Therefore,
sum of interior angles = 3 × 360° = 1080°

Again, we
have sum of interior angles, S = (n - 2)180°, where n is the number of sides of
the polygon

⇒ (n - 2)180° = 1080°⇒ n - 2 = 6

⇒ n = 8

4. Q. (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

Answer: The polygon with minimum number of sides is a triangle, and each angle of an equilateral triangle measures 60°, so 60° is the minimum value of the possible interior angle for a regular polygon. For an equilateral triangle the exterior angle is 180°-60°=120° and this is the maximum possible value of an exterior angle for a regular polygon.

The sum of the exterior angles of any polygon= 360

^{0}

Hence, the polygon of 8 sides is octagon.

5.
Q. Find the measure of each exterior angle of a regular polygon of 9 sides.

Ans:
Total measure of all exterior angles
= 360

No. of sides = 9

Measure of each exterior angle =
360/9 = 40

6.Q.
If the sum of the measures of the interior angles of a polygon equals the sum
of the measures of the exterior angles, how many sides does the polygon have?

Ans:The sum
of the measures of the interior angles
of a polygon with n sides =(n-2)x180

^{0}
(n-2) x180

^{0 }= 360^{0 }Þ n=2+2=4
7.Q. The sum of the interior angles of a regular polygon is:

**(n - 2) × 180°**where**n**is the number of sides of the polygon.
Solution: The sum of its exterior angles of regular polygon= 360°

The exterior angle of a regular polygon

Interior angle of a regular polygon = sum of interior angles ÷ number of
sides

8. Q.What is the measure of the each angle of regular
Hexagon?

Ans: No. of sides in regular hexagon = 6

The measure of the each angle =[(2n – 4)x90

^{0 }/n ]=[2x6-4]x90^{0}/6 =720^{0 }/6 =120^{0}
9. Q. Find the number of sides of a polygon whose each interior angle is 156

^{0}.
Ans each exterior
angle = 180 - 1560 = 24

^{0}
Measure of
Each Exterior Angle of a Polygon = 360/n

^{0 }= 360/n Þ n = 360/24 =15

10.Q.
Two
regular polygons are such that the ratio between their no. of sides is 1:2 and
the ratio of measures of their interior angle is 3:4. Find the number of sides
of each polygon.

Ans: let the number of sides are x and 2x

then their interior angles will be [{(2n-4)/n}x90

^{0}] and [{(4n-4)/n}x90^{0}]
A/Q, the ratio of measures of their interior angle = 3:4

Þ [{(2n-4)/n}x90

^{0}] ¸ [{(4n-4)/n}x90^{0}] = ¾
On solving this we get , n=5

So, the numbers of sides are 5 and 2x5=10

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