Question

The ratio between exterior angle and interior angle of a
regular polygon is 1:3. Find the number sides of the polygon. Also find each exterior and intirior angles​​

Answer 1

Answer:

8 sides

Step-by-step explanation:

The ratio between exterior angle and interior angle of a  regular polygon is 1:3

exterior angle = x

interior angle = 3x

x + 3x = 180 degree

4x =  180

x = 45 degree

sum of exterior angles of a regular polygon = 360 degree

45 * number of sides = 360

number of sides = 360/45

number of sides of the regular polygon = 8

Answer 2

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✺Answer:

♦️GiveN:

• Ratio of exterior to interior angle is 1:3

♦️To FinD

• Number of sides
• Exterior angle
• Interior angle

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✺Explanation of Q.

▶️ Refer to the attachment

Whatever the number of sides be, the interior and exterior angles forms a linear pair of angles. Hence there sum would be 180°. According to question, we have their ratio and sum i.e. 180. So we can find the value of each angle.

Now, it is given that, the polygon is regular, so each of the angle of polygon is equal. We know, Number of sides = Number of angles. So, simply dividing the Total sum of exterior angles by measure of each exterior angle gives us the no. of sides of the polygon.

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✺Concept Used:

We know that Sum of all Exterior angles of any polygon of n sides is 360° . So, after finding out the value of each Exterior angle, will give the number of sides.

$\large{ \rm{ \rightarrow \: sum \: of \: all \: exterior \: angles = 360}} \\ \\ \large{ \rm{ \rightarrow \: let \: each \: exterior \: angle \: be \: = a}} \\ \\ \large{\boxed{ \green{\rm{ \rightarrow \: no. \: of \: sides = \frac{360}{a} }}}}$

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✺Solution:

Let the exterior angle be x

Then, Interior angle will be 3x

We know, x + 3x = 180 (linear pair)

Here, x is exterior angle and 3x is interior.

$\large{ \rm{ \rightarrow \: x + 3x = 180}} \\ \\ \large{ \rm{ \rightarrow \: 4x = 180}} \\ \\ \large{ \rm{ \rightarrow \: x = \cancel{\frac{180}{4}} = 45}} \\ \\ \large{ \rm{ \pink{ \therefore{exterior \: angle = x = 45}}}} \\ \\ \large{ \rm{ \pink{ \therefore{interior \: angle = 3x = 135}}}}$

Finding No. of sides by using formula,

$\large{ \rm{ \rightarrow \: no. \: of \: sides = \frac{sum \: of \: exterior \: angles}{measure \: of \: each \: exterior \: angle}}} \\ \\ \large{ \rm{ \rightarrow \: n = \frac{360}{45} = 8} }\\ \\ \large{\rm{\pink{\therefore {number \:of\: sides \:= \:8}}}}$

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