Below is a picture of Devils Postpile, near Mammoth Lakes, California. These posts are cooled lava (called columnar basalt) and as the lava pools and cools, it ideally would form regular hexagonal columns. However, variations in cooling caused some columns to either not be perfect or pentagonal.
First, define regular in your own words. Then, what is the sum of the angles in a regular hexagon? What would each angle be? After completing this Concept you'll be able to answer questions like these.
Watch This
CK-12 Foundation: Chapter6InteriorAnglesinConvexPolygonsA
Watch the first half of this video.
James Sousa: Angles of Convex Polygons
Guidance
Recall that interior angles are the angles inside a closed figure with straight sides. As you can see in the images below, a polygon has the same number of interior angles as it does sides.
A diagonal connects two non-adjacent vertices of a convex polygon. Also, recall that the sum of the angles in a triangle is \begin{align*}180^\circ\end{align*}. What about other polygons?
Investigation: Polygon Sum Formula
Tools Needed: paper, pencil, ruler, colored pencils (optional)
1. Draw a quadrilateral, pentagon, and hexagon.
2. Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles.
Make sure none of the triangles overlap.
3. Make a table with the information below.
Name of Polygon | Number of Sides | Number of \begin{align*}\triangle s\end{align*} from one vertex | (Column 3) \begin{align*}\times\end{align*} (\begin{align*}^\circ\end{align*} in a \begin{align*}\triangle\end{align*}) | Total Number of Degrees |
---|---|---|---|---|
Quadrilateral | 4 | 2 | \begin{align*}2 \times 180^\circ\end{align*} | \begin{align*}360^\circ\end{align*} |
Pentagon | 5 | 3 | \begin{align*}3 \times 180^\circ\end{align*} | \begin{align*}540^\circ\end{align*} |
Hexagon | 6 | 4 | \begin{align*}4 \times 180^\circ\end{align*} | \begin{align*}720^\circ\end{align*} |
4. Do you see a pattern? Notice that the total number of degrees goes up by \begin{align*}180^\circ\end{align*}. So, if the number sides is \begin{align*}n\end{align*}, then the number of triangles from one vertex is \begin{align*}n - 2\end{align*}. Therefore, the formula would be \begin{align*}(n - 2) \times 180^\circ\end{align*}.
Polygon Sum Formula: For any \begin{align*}n-\end{align*}gon, the sum of the interior angles is \begin{align*}(n - 2) \times 180^\circ\end{align*}.
A regular polygon is a polygon where all sides are congruent and all interior angles are congruent.
Regular Polygon Formula: For any equiangular \begin{align*}n-\end{align*}gon, the measure of each angle is \begin{align*}\frac{(n-2)\times 180^\circ}{n}\end{align*}.
Example A
Find the sum of the interior angles of an octagon.
Use the Polygon Sum Formula and set \begin{align*}n = 8\end{align*}.
\begin{align*}(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ\end{align*}
Example B
The sum of the interior angles of a polygon is \begin{align*}1980^\circ\end{align*}. How many sides does this polygon have?
Use the Polygon Sum Formula and solve for \begin{align*}n\end{align*}.
\begin{align*}(n - 2) \times 180^\circ & = 1980^\circ\\ 180^\circ n - 360^\circ & = 1980^\circ\\ 180^\circ n & = 2340^\circ\\ n & = 13 \qquad \text{The polygon has} \ 13 \ \text{sides.}\end{align*}
Example C
How many degrees does each angle in an equiangular nonagon have?
First we need to find the sum of the interior angles in a nonagon, set \begin{align*}n = 9\end{align*}.
\begin{align*}(9 - 2) \times 180^\circ = 7 \times 180^\circ = 1260^\circ\end{align*}
Second, because the nonagon is equiangular, every angle is equal. Dividing \begin{align*}1260^\circ\end{align*} by 9 we get each angle is \begin{align*}140^\circ\end{align*}.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter6InteriorAnglesinConvexPolygonsB
Concept Problem Revisited
A regular polygon has congruent sides and angles. A regular hexagon has \begin{align*}(6-2)180^\circ=4\cdot180^\circ=720^\circ\end{align*} total degrees. Each angle would be \begin{align*}720^\circ\end{align*} divided by 6 or \begin{align*}120^\circ\end{align*}.
Vocabulary
The interior angle of a polygon is one of the angles on the inside. A regular polygon is a polygon that is equilateral (has all congruent sides) and equiangular (has all congruent angles).
Guided Practice
1. Find the measure of \begin{align*}x\end{align*}.
2. The interior angles of a pentagon are \begin{align*}x^\circ, x^\circ, 2x^\circ, 2x^\circ,\end{align*} and \begin{align*}2x^\circ\end{align*}. What is \begin{align*}x\end{align*}?
3. What is the sum of the interior angles in a 100-gon?
Answers:
1. From the Polygon Sum Formula we know that a quadrilateral has interior angles that sum to \begin{align*}(4-2) \times 180^\circ=360^\circ\end{align*}.
Write an equation and solve for \begin{align*}x\end{align*}.
\begin{align*}89^\circ + (5x - 8)^\circ + (3x + 4)^\circ + 51^\circ & = 360^\circ\\ 8x & = 224\\ x & = 28\end{align*}
2. From the Polygon Sum Formula we know that a pentagon has interior angles that sum to \begin{align*}(5-2) \times 180^\circ=540^\circ\end{align*}.
Write an equation and solve for \begin{align*}x\end{align*}.
\begin{align*} x^\circ + x^\circ + 2x^\circ + 2x^\circ + 2x^\circ&=540^\circ\\ 8x&=540\\x&=67.5\end{align*}
3. Use the Polygon Sum Formula. \begin{align*}(100-2) \times 180^\circ=17,640^\circ\end{align*}.
Interactive Practice
Practice
- Fill in the table.
# of sides | Sum of the Interior Angles | Measure of Each Interior Angle in a Regular \begin{align*}n-\end{align*}gon | |
---|---|---|---|
3 | \begin{align*}60^\circ\end{align*} | ||
4 | \begin{align*}360^\circ\end{align*} | ||
5 | \begin{align*}540^\circ\end{align*} | \begin{align*}108^\circ\end{align*} | |
6 | \begin{align*}120^\circ\end{align*} | ||
7 | |||
8 | |||
9 | |||
10 | |||
11 | |||
12 |
- What is the sum of the angles in a 15-gon?
- What is the sum of the angles in a 23-gon?
- The sum of the interior angles of a polygon is \begin{align*}4320^\circ\end{align*}. How many sides does the polygon have?
- The sum of the interior angles of a polygon is \begin{align*}3240^\circ\end{align*}. How many sides does the polygon have?
- What is the measure of each angle in a regular 16-gon?
- What is the measure of each angle in an equiangular 24-gon?
- Each interior angle in a regular polygon is \begin{align*}156^\circ\end{align*}. How many sides does it have?
- Each interior angle in an equiangular polygon is \begin{align*}90^\circ\end{align*}. How many sides does it have?
For questions 10-18, find the value of the missing variable(s).
- The interior angles of a hexagon are \begin{align*}x^\circ, (x + 1)^\circ, (x + 2)^\circ, (x + 3)^\circ, (x + 4)^\circ,\end{align*} and \begin{align*}(x + 5)^\circ.\end{align*} What is \begin{align*}x\end{align*}?