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प्रश्न
We know the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
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उत्तर
Since, the sum of all interior angles of a polygon of n sides
= (2n – 4) × 90°
∴ Sum of interior angles of a polygon of sides 3
= (2 × 3 – 4) × 90°
= 180°
Sum of interior angles of a polygon of sides 4
= (2 × 4 – 4) × 90°
= 360°
Similarly, the sum of interior angles of the polygon of sides,
5, 6, 7 … are 540°, 720°, 900°…
Therefore, the series will be 180°, 360°, 540°, 720°, 900°… which is A.P.
Here a = 180°
d = 180°
We have to find the sum of interior angles of a polygon of 21 sides
i.e. 19th term
an = a + (n – 1)d
a19 = 180° + (19 – 1)180°
= 180° + 18 × 180°
= 180° + 3240°
= 3420°
Hence, the required sum of interior angles = 3420°.
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