Question

What is the ratio of the interior angle of a pentagon and a decagon.

Answer 1

Pentagon = 5 sided polygon

Decagon = 10 sided polygon

.

$\boxed {\text{Interior angle = }\dfrac{(n-2) \times 180 }{n} }$

.

Find the interior angle of a pentagon:

$\text{Interior angle of a pentagon = }\dfrac{(5-2) \times 180 }{5} } = 108 \textdegree$

.

Find the interior angle of a decagon:

$\text{Interior angle of a decagon = }\dfrac{(10-2) \times 180 }{10} } = 144 \textdegree$

.

Ratio of the angles:

Pentagon : Decagon = 108 : 144 = 3 :4

.

Answer: The ratio is 3:4

Answer 2

Answer:

To Find:-

Ratio of the interior angle of a Pentagon and a Decagon.

We know,

Interior Angle of a polygon of side n:-

$\boxed{\large\tt\blue{\frac {(n-2) \times {180}^{\circ}}{n}}}$

Therefore,

Interior angle of a Pentagon:-

Sides of a Pentagon = 5

Hence as per the formula, expression will be:-

$\tt{\frac {(5-2) \times {180}^{\circ}}{5}}$

(Expression as per the formula given above)

$\tt{= \frac {3 \times {180}^{\circ}}{5}}$

(In this step, we have Subtracted 2 from 5 and the result is 3. This change can be seen in the nominator)

$\tt{= \frac {3 \times {\cancel {{180}^{\circ}}}}{{\cancel {5}}}}$

(In this step, we have cancelled 180° with 5 taking common 5)

$\tt{= \frac {3 \times {36}^{\circ}}{1}}$

(Result after Dividing them)

$\tt\purple{= {108}^{\circ}}$

(After Multiplication/Answer)

Therefore, interior angle of a Pentagon is 108°.

NOW,

Interior angle of a Decagon:-

Sides of a Decagon = 10

Hence as per the formula, expression will be:-

$\tt{\frac {(10-2) \times {180}^{\circ}}{10}}$

(Expression as per the formula given above)

$\tt{= \frac {8 \times {180}^{\circ}}{10}}$

(In this step, we have Subtracted 2 from 10 and the result is 8, this change can be seen in the nominator)

$\tt{= \frac {8 \times {\cancel {{180}^{\circ}}}}{{\cancel {10}}}}$

(In this step, we have cancelled 180° with 10 taking common 10)

$\tt{= \frac {8 \times {18}^{\circ}}{1}}$

(Result after Dividing them)

$\tt\purple{= {144}^{\circ}}$

(After Multiplication/Answer)

_____________...

Hence, ratio between their angles:-

$\tt{Pentagon:Decagon}$

(Written in this way so that we can get the suitable answer)

$\tt{= \frac {Pentagon}{Decagon}}$

(As we know, numbers in ratio (:) form equals to division (÷) form)

$\tt{= \frac {{108}^{\circ}}{{144}^{\circ}}}$

(Putting their values to their respective places)

$\tt{= \frac {{\cancel {{108}^{\circ}}}}{{\cancel {{144}^{\circ}}}}}$

(Cancelled)

$\tt{= \frac {3}{4}}$

(Simplest form)

$\boxed{\huge\tt\green{=3:4}}$

$\boxed{\rm{(ANSWER)/Ratio}}$

_______...

REQUIRED ANSWER:-

$\tt{\therefore}$ Ratio of interior angles of a Pentagon and a Decagon is $\boxed{\large\tt\orange{3:4}}$.