ACT Math - How to find an angle in a polygon

Find the total number of degrees inside a hexagon.

Explanation

To solve, simply use the following formula where is the number of sides. Thus,

$\textup{degrees}=(n-2)(180)=(6-2)(180)=4*180=720$

In the diagram below, what is $\theta$ equal to?

Hexagon_sides

$\tan^{-1}\left ( \frac{s}{r} \right )$

$\tan^{-1}\left ( \frac{r}{s} \right )$

$\sin^{-1}\left ( \frac{r}{s} \right )$

$\tan \left ( \frac{r}{s} \right )$

$\tan \left ( \frac{s}{r} \right )$

Explanation

The figure given is a hexagon with an embedded triangle. The fact that it is embedded in a triangle is mainly to throw you off, as it has little to no consequence on the correct answer. Of the available answer choices, you must choose a relationship that would give the value of $\theta$ . Tangent describes the relationship between an angle and the opposite and adjacent sides of that angle. Or in other words, tan $\theta$ = opposite side/adjacent side. However, when solving for an angle, we must use the inverse function. Therefore, if we know the opposite and adjacent sides are, we can use the inverse of the tangent, or arctangent (tan-1), of $\frac{s}{r}$ to find $\theta$ .

Thus,

$\theta =\tan^{-1}\left ( \frac{s}{r} \right )$

What is the total number of degrees in a $\textup{12-sided}$ polygon?

$1800^{\circ}$

$1440^{\circ}$

$2160^{\circ}$

$150^{\circ}$

$144^{\circ}$

Explanation

To find the total number of degrees in an -sided polygon, use the formula:

thus we see that $1800^{\circ}$

Shape_1

What is the value of angle in the figure above?

$100^{\circ}$

$50^{\circ}$

$35^{\circ}$

$110^{\circ}$

$75^{\circ}$

Explanation

Begin by noticing that the upper-right angle of this figure is supplementary to $70^{\circ}$ . This means that it is $180^{\circ}-70^{\circ}=110^{\circ}$ :

Shape_1

Now, a quadrilateral has a total of $360^{\circ}$ . This is computed by the formula , where represents the number of sides. Thus, we know:

$110^{\circ}+70^{\circ}+80^{\circ}+x=360^{\circ}$

This is the same as $260^{\circ} + x = 360^{\circ}$

Solving for , we get:

$x=100^{\circ}$

Fig_2

What is the angle measure for the largest unknown angle in the figure above? Round to the nearest hundredth.

$186.36^{\circ}$

$74.55^{\circ}$

$149.1^{\circ}$

$84.14^{\circ}$

$140.14^{\circ}$

Explanation

The total degree measure of a given figure is given by the equation , where represents the number of sides in the figure. For this figure, it is: $180*(6-2)=180*4 = 720^{\circ}$

Therefore, we know that the sum of the angles must equal $720^{\circ}$ . This gives us the equation:

$90^{\circ}+120^{\circ}+100^{\circ}+x+2.5x+2x=720^{\circ}$

Simplifying, this is:

$310^{\circ}+5.5x=720^{\circ}$

Now, just solve for :

$x=\frac{410}{5.5}$

The largest of the unknown angles is or $2.5 * \frac{410}{5.5}=186.36363636363636...^{\circ}$

Rounding, this is $186.36^{\circ}$ .

Find the total number of degrees in a heptagon.

$900^\circ$

$720^\circ$

$1080^\circ$

$540^\circ$

Explanation

To solve, simply use the formula for finding the degrees in a closed polygon, given n being the number of sides.

In this particular case, a heptagon has seven sides.

Thus,

What is the interior angle of a $\textup{9-sided}$ polygon (a nonagon)? Round answer to the nearest hundredth if necessary.

$140^{\circ}$

$1260^{\circ}$

$147.27^{\circ}$

$128.57^{\circ}$

$360^{\circ}$

Explanation

To find the interior angle of an sided polygon, first find the total number of degrees in the polygon by the formula:
$(n-2)*180^{\circ}$ . For us that yields:
$(9-2)*180^{\circ} = 1260^{\circ}$ . Next we divide the total number of degrees by the number of sides: