Question

The angle measures of an octagon are in
Arithmetic sequence.
a) What is the sum of its angles?
b) What is the sum of its smallest and
largest angles?

Answer 1

Step-by-step explanation:

An octagon has eight sides, so the sum of the angles of the octagon is 180(8 – 2) = 180(6) = 1080 degrees. Because the octagon is regular, all of its sides and angles are congruent. Thus, the measure of each angle is equal to the sum of its angles divided by 8.

Answer 2

Answer:

(a) 1080°

(b) 270°

Step-by-step explanation:

(a) Sum of the angles in a polygon with n sides is given by, (n – 2) × 180°

For octagon,

sides = 8

Sum of the angles

= (8 – 2) × 180°

= 6 × 180°

= 1080°

(b) The angle measures of an octagon are in A.P.

Let the first angle be 'a'.

nth term of an A.P is given by,

$\bf a_n = a + (n-1)d$

Largest angle will be 8th angle.

a8 = a + (8 – 1)d = a + 7d

Sum of n terms is given by,

$\sf S_n = \dfrac{n}{2}[2a+(n-1)d]$

Sum of all the angles is 1080°

Put n = 8,

1080 = 8/2 [2a + (8 – 1)d]

1080 = 4 [ 2a + 7d ]

2a + 7d = 1080/4

2a + 7d = 270

a + a + 7d = 270

a is the smallest angle

a + 7d is the largest angle

Hence, sum of smallest and largest angles is 270°.