Question

date: __________ per: ________ unit 7: polygons & quadrilaterals homework 1: angles of polygons this is a 2 - page document! 1. what is the sum of the measures of the interior angles of an octagon? ________ 2. what is the sum of the measures of the interior angles of a 25 - gon? 180° 3. what is the measure of each interior angle of a regular hexagon? 120° 4. what is the measure of each interior angle of a regular 16 - gon? ________ 5. what is the sum of the measures of the exterior angles of a decagon? ________ 6. what is the measure of each exterior angle of a regular 30 - gon? ________ 7. an exterior angle of a regular polygon measures 22.5°. how many sides does it have? ________ 8. an interior angle of a regular polygon measures 170°. how many sides does it have? __________

Explanation:

Step1: Recall interior - angle sum formula

The sum of the interior angles of a polygon is given by $S=(n - 2)\times180^{\circ}$, where $n$ is the number of sides of the polygon.

Step2: Solve for octagon

For an octagon, $n = 8$. Then $S=(8 - 2)\times180^{\circ}=6\times180^{\circ}=1080^{\circ}$.

Step3: Solve for 25 - gon

For a 25 - gon, $n = 25$. Then $S=(25 - 2)\times180^{\circ}=23\times180^{\circ}=4140^{\circ}$.

Step4: Recall measure of each interior angle formula for regular polygon

The measure of each interior angle $A$ of a regular polygon is $A=\frac{(n - 2)\times180^{\circ}}{n}$.

Step5: Solve for regular hexagon

For a regular hexagon, $n = 6$. Then $A=\frac{(6 - 2)\times180^{\circ}}{6}=\frac{4\times180^{\circ}}{6}=120^{\circ}$.

Step6: Solve for regular 16 - gon

For a regular 16 - gon, $n = 16$. Then $A=\frac{(16 - 2)\times180^{\circ}}{16}=\frac{14\times180^{\circ}}{16}=157.5^{\circ}$.

Step7: Recall exterior - angle sum property

The sum of the exterior angles of any polygon is always $360^{\circ}$. So for a decagon, the sum of the exterior angles is $360^{\circ}$.

Step8: Solve for each exterior angle of regular 30 - gon

The measure of each exterior angle $E$ of a regular polygon is $E=\frac{360^{\circ}}{n}$. For a regular 30 - gon, $n = 30$, so $E=\frac{360^{\circ}}{30}=12^{\circ}$.

Step9: Find number of sides from exterior angle

If the exterior angle $E = 22.5^{\circ}$, and $E=\frac{360^{\circ}}{n}$, then $n=\frac{360^{\circ}}{22.5^{\circ}} = 16$.

Step10: Find number of sides from interior angle

If the interior angle $A = 170^{\circ}$, first find the exterior angle $E=180^{\circ}-A = 180^{\circ}-170^{\circ}=10^{\circ}$. Since $E=\frac{360^{\circ}}{n}$, then $n=\frac{360^{\circ}}{10^{\circ}}=36$.

Answer:

1. $1080^{\circ}$
2. $4140^{\circ}$
3. $120^{\circ}$
4. $157.5^{\circ}$
5. $360^{\circ}$
6. $12^{\circ}$
7. $16$
8. $36$