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 hexagonal.
First, define regular in your own words. Then, what is the sum of the angles in a regular hexagon? What would each angle be?
Interior Angles in Convex Polygons
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*}.
Finding the Sum of Interior Angles
1. 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*}
2. 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*}
Watch the first half of this video.
Measuring Angles
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*}.
Earlier 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*}.
Examples
Example 1
Find the measure of \begin{align*}x\end{align*}.
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*}
Example 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*}?
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*}
Example 3
What is the sum of the interior angles in a 100-gon?
Use the Polygon Sum Formula. \begin{align*}(100-2) \times 180^\circ=17,640^\circ\end{align*}.
Review
- 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*}?
Review (Answers)
To view the Review answers, open this PDF file and look for section 6.1.