Question

What is the measure of an interior angle 25-gon ?

Answer 1

Hint: To simplify this question , we need to solve it step by step . In order to solve the given question , we should know the important concepts related to the question that the measure of each pair of exterior and interior angles of a polygon adds upto \[{180^ \circ }\]. Also the fact that we are going to use is the sum of measures of all the exterior angles of any polygon , irrespective of its number of sides is always \[{360^ \circ }\].

Complete step-by-step solution:
We are given the question that finds the measure of an interior angle of 25 sided regular polygon .
As we know the two properties of a polygon that are as follows –
>The measure of each pair of exterior and interior angles of a polygon adds up to \[{180^ \circ }\].
>The sum of measures of all the exterior angles of any polygon , irrespective of its number of sides is always \[{360^ \circ }\].
Sum of the measures of all the exterior angles and all interior angles of a 25-sided polygon is
\[25 \times {180^ \circ } = {4500^ \circ }\].
The sum of measures of all the exterior angles of any polygon , irrespective of its number of sides is always \[{360^ \circ }\]. So, we are going to subtract the measure of all exterior angles from the sum of the measures of all the exterior angles and all interior angles of a 25-sided polygon so that we can get the interior angles only which can be calculated as –
${4500^ \circ } - {360^ \circ } = {4140^ \circ }$
Hence, the sum of all the interior angles of a 25-sided polygon is ${4140^ \circ }$.
Now , to find the measure of each interior angle of a regular 25-sided polygon we will divide the sum of all interior angles of a 25-sided polygon by the number of sides -25 .
 The measure of each interior angle of a regular 25-sided polygon =$\dfrac{{{{4140}^ \circ }}}{{25}} = {165.6^ \circ }$.
Therefore , the required answer is ${165.6^ \circ }$.
Additional Information :
From the above information, we can see that for an n sided polygon
The measure of each interior angles of n sided polygon can be solved by the formula = \[
   = \dfrac{{{{180}^ \circ }(n - 2)}}{n} \\
   = \dfrac{{{{180}^ \circ }(25 - 2)}}{{25}} \\
   = \dfrac{{{{4140}^ \circ }}}{{25}} \\
   = {165.6^ \circ } \\
 \]
Sum of the measures of the interior angles of a 25 sided polygon is \[{4140^ \circ }\]

Note: Always try to understand the mathematical statement carefully and keep things distinct . Remember the properties and apply appropriately. Choose the options wisely , it's better to break the question and then solve part by part. Cross check the answer and always keep the final answer simplified .