The 4 angles of a quadrilateral add to 360

The 5 angles of a pentagon add to 540

The 6 angles of a hexagon add to 720

To solve, simply use the formula for the total degrees in a polygon, where n is the number of vertices.

In this particular case, a nonagon is a shape with nine sides and thus nine vertices.

Thus,

If one exterior angle is taken at each vertex of any convex polygon, the sum of their measures is . In a regular polygon - one with congruent sides and congruent interior angles, each exterior angle is congruent to one another. If the polygon has  sides, each exterior angle has measure .

Given the common measure , 

Multiplying both sides by :

and

Since  is equal to a number of sides, it is a whole number. Thus, we are looking for a value of  which, when we divide 360 by it, yields a non-whole result. We see that  is the correct choice, since'

A quick check confirms that 360 divided by 8, 10, 12, or 15 yields a whole result.

The measure of each interior angle of an -sided polygon can be calculated using the formula

Setting :

The correct choice is therefore .

The easiest way to work this is arguably to examine the exterior angles, each of which forms a linear pair with an interior angle. If an interior angle measures , then each exterior angle, which is supplementary to an interior angle, measures

The measures of the exterior angles of a polygon, one per vertex, total ; in a regular polygon, they are congruent, so if there are  such angles, each measures . Since the number of vertices is equal to the number of sides, if we set this equal to  and solve for , we will find the number of sides.

Multiply both sides by :

The polygon has 72 vertices and, thus, 72 sides.

To determine the measure of the angles of a regular polygon use:

Angle = (n – 2) x 180° / n

Thus, (n – 2) x 180° / n = 140°

180° n - 360° = 140° n

40° n = 360°

n = 360° / 40° = 9

The formula for of interior angles based on a polygon with a number of side n is:

Each Interior  Angle = (n-2)*180/n

= (7-2)*180/7 = 128.57 degrees

We need to find the area of both the square and the circle and then subtract the two.  Inscribed means draw within a figure so as to touch in as many places as possible.  So the circle is drawn inside the square.  The opposite is circumscribed, meaning drawn outside.

A_square = s² = 36 cm² so the side is 6 cm

6 cm is also the diameter of the circle and thus the radius is 3 cm

A circle = πr² = 3.14 * 3² = 28.28 cm²

The resulting difference is 7.74 cm²

We can find the area of the shaded region by subtracting the area of the semicircles, which is much easier to find. Two semi-circles are equivalent to one full circle. Thus we can just use the area formula, where r = 6:

π(6²) → 36π

Now we must subtract the area of the semi-circles from the total area of the square. Since we know that the radius also covers half of a side, 6(2) = 12 is the full length of a side of the square. Squaring this, 12² = 144. Subtracting the area of the circles, we get our final terms,

= 144 – 36π

First find the area of both squares using the formula .

For square A, s = 5.

For square B, s = 25.

The question is asking for the ratio of these two areas, which will tell us how many times bigger square B is. Divide the area of square B by the area of square A to find the answer.