Question

3. Find the measure of the each interior angle of a regular polygon of:
(b) 8 sides
(c) 15 sides
(a) 6 sides

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Answer 1

Required Solution :

How to solve?

To solve this, we know the formula to measure the each interior angle of a regular polygon:

For a regular polygon,

★ $\boxed {\sf{ Each \: interior \: angle ={ 180 }^{\circ} – (each \: exterior \: angle) }}$

So, in order to calculate measure of the each interior angle , we have to also calculate its each exterior angle :

★ $\boxed {\sf{ Each \: exterior \: angle = {\Bigg\lgroup \dfrac{360}{n} \Bigg \rgroup}^{\circ} }}$

[n = number of sides]

Now, simply we can say that :

★ $\boxed {\sf \red{ Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{n} \Bigg \rgroup}^{\circ} }}$

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Solution:

(a) 6 sides

By using the formula :-

★ $\boxed {\sf \red{ Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{n} \Bigg \rgroup}^{\circ} }}$

$\sf { \implies Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{6} \Bigg \rgroup}^{\circ} }$

$\sf { \implies Each \: interior \: angle = {180}^{\circ} - {60}^{\circ} }$

$\bf { \implies Each \: interior \: angle = {120}^{\circ} }$

Hence, the measure of the each interior angle of a regular polygon of 6 sides is 120°.

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(b) 8 sides

By using the formula :-

★ $\boxed {\sf \red{ Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{n} \Bigg \rgroup}^{\circ} }}$

$\sf { \implies Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{8} \Bigg \rgroup}^{\circ} }$

$\sf { \implies Each \: interior \: angle = {180}^{\circ} - {45}^{\circ} }$

$\bf { \implies Each \: interior \: angle = {135}^{\circ} }$

Hence, the measure of the each interior angle of a regular polygon of 8 sides is 135°.

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(c) 15 sides

By using the formula :-

★ $\boxed {\sf \red{ Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{n} \Bigg \rgroup}^{\circ} }}$

$\sf { \implies Each \: interior \: angle = {180}^{\circ} - {\Bigg\lgroup \dfrac{360}{15} \Bigg \rgroup}^{\circ} }$

$\sf { \implies Each \: interior \: angle = {180}^{\circ} - {24}^{\circ} }$

$\bf { \implies Each \: interior \: angle = {156}^{\circ} }$

Hence, the measure of the each interior angle of a regular polygon of 15 sides is 156°.

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