Peculiar polygon

We know that if the number of sides of a polygon is $n$ , then the sum of interior angles of the polygon will be equal to $(n-2)\times180^\circ$ .

Therefore, the sum of interior angles of a pentagon will be $(5-2)\times 180^\circ = 540^\circ$ .

Let us assume the measure of the smallest interior angle to be $x$ . Therefore, the other angles would be $2x$ , $3x$ , $4x$ , $8x$ . Therefore the sum of all angles would be:

$x + 2x + 3x + 4x + 8x = 540^\circ\\ 18x = 540^\circ \implies \boxed{x = 30^\circ}$

The "solvers" all use appropriate analysis to conclude that the answer must be 30, but they fail to test the reasonableness of the answer against the stated problem. We are given a CONVEX polygon, so all interior angles must be less than 180 degrees. If the smallest angle is 30 degrees, then the largest angle must be 240 degrees, which violates a premise of the problem. The conditions stated in the problem cannot be met.

There is no need to apply a formula if we can imagine geometric figures.

Note that a five sided polygon , that is pentagon is nothing , but a quadrilateral joined with a triangle so that they have an edge common. So sum of all interior angles of pentagon $=180^\circ +360 ^\circ = 540^\circ$ .

Let the smallest interior angle be $x$ and eventually according to given condition in problem , other angles are $2x,3x,4x,8x$ and as shown above , their sum is $540^\circ$ . Hence we form the equation:

$x+2x+3x+4x+8x= 540^\circ \Rightarrow 18x=540^\circ \Rightarrow x=\dfrac{540^\circ}{18} \Rightarrow \boxed{x=30^\circ}$

The sum of interior angles is $540$ . We let $x$ be the smallest angle, then

$540=2x+3x+4x+8x+x$

$x=30$

Lets call the smallest angle A, then B, then C and D and E in order of lest to greatest measure. Angle A=x , Angle B=2x , Angle C=3x, Angle D=4x, Angle E=8x. Knowing that the sum of the interior angles of all pentagons is 540 degrees, (5-2)(180), since the sum of the interior angles of any n-gon is (n-2)(180). So you get 540=x+2x+3x+4x+8x, the measures of all the interior angles and also substituting their values. Adding like terms, you get 540=18x, divide and you get x=30, remember that the smallest angle equals x so the answer is 30