Angles of Polygon Game
To find the interior angles of a polygon, you first need to know how many sides it has, and whether it’s a regular or irregular polygon.
Scroll down the page for a more detailed explanation.
This new game will focus on finding the interior angles of polygons with three modes: calculating the sum of interior angles, finding the size of one interior angle in a regular polygon, and a mixed mode.
 

Polygon Angles Master

Score: 0 Question: 0

Select Challenge Mode

Next Polygon ➜ ← Back to Menu

Choose your geometry challenge:

Sum of Interior Angles One Interior Angle (Regular) Mixed Problems

 

How to Play the Angles of Polygon Game
Here’s how to play:

1. Start: Each Quiz consists of 10 questions. Select the difficulty: Sum of Interior Angles, One Interior Angle (Regular), or Mixed Problems.
   
2. Look at the Problem: Use either the Sum of Interior Angles formula or Single Interior Angle formula.
   
3. Select Your Answer: Select the correct fraction.
   
4. Check Your Work: If you selected the right answer, it will be highlighted in green. If you are wrong, it will be highlighted in red and the correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Polygon” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, click “Finish Game” to get your final score.
    
   

Finding the Interior Angles of a Polygon
The process for finding the interior angles of a polygon depends on whether you are looking for the total sum of all angles or the measure of a single angle, and if the polygon is regular (all sides and angles are equal) or irregular (sides and angles vary).
Finding the Sum of Interior Angles (Any Polygon)
The total measure of all interior angles in any convex polygon is determined only by the number of sides it has. This formula works for both regular and irregular polygons.
Formula for the Sum of Interior Angles (S)
If n is the number of sides (or vertices) of the polygon, the sum of its interior angles (S) is given by:
\(\Large S = (n - 2) \times 180^{\circ}\)

Example
Problem: Find the sum of the interior angles of an octagon.
Determine n: An octagon has 8 sides, so n = 8.
Apply the formula:
\(S = (8 - 2) \times 180^{\circ}\)
\(S = 6 \times 180^{\circ}\)
\(S = 1080^{\circ}\)
The sum of the interior angles of any octagon is \(1080^{\circ}\).

Finding a Single Interior Angle (Regular Polygon Only)
A regular polygon has all sides equal in length and all interior angles equal in measure. If you know the total sum, you can simply divide by the number of angles (n).
Formula for a Single Interior Angle (A)
For a regular polygon with n sides, the measure of one interior angle (A) is:
\(\Large A = \frac{(n - 2) \times 180^{\circ}}{n}\)

Example
Problem: Find the measure of one interior angle of a regular pentagon.
Determine n: A pentagon has 5 sides, so n = 5.
Apply the formula:
\(A = \frac{(5 - 2) \times 180^{\circ}}{5}\)
\(A = \frac{3 \times 180^{\circ}}{5}\)
\(A = \frac{540^{\circ}}{5}\)
\(A = 108^{\circ}\)
Each interior angle in a regular pentagon measures \(108^{\circ}\).

Finding Interior Angles (Irregular Polygon)
For an irregular polygon (where angles are not equal), you must first find the sum of the angles using the formula \(S = (n - 2) \times 180^{\circ}\).
Once you have the sum (S):
You need n-1 of the n angles given.
Subtract the known angles from the total sum ($S$) to find the measure of the final, unknown angle.

Example
Problem: An irregular quadrilateral (4 sides) has three interior angles measuring 70°, 110°, and 95°. Find the fourth angle (\(x\)).
Find the Sum (S):
\(S = (4 - 2) \times 180^{\circ} = 2 \times 180^{\circ} = 360^{\circ}\)
Subtract the known angles from the sum:
\(x = 360^{\circ} - (70^{\circ} + 110^{\circ} + 95^{\circ})\)
\(x = 360^{\circ} - 275^{\circ}\)
\(x = 85^{\circ}\)
The fourth interior angle measures \(85^{\circ}\).

This video gives a clear, step-by-step approach to learn how to use the Angles of Polygon Formula.

• Show Video

 

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.