Sum of Interior Angles in a Polygon
A student-based discovery activity that explores the sum of the interior angles of a polygon by deconstructing the polygons into triangles, and then calculating the sum of degrees for every triangle that could be made.
Tabla de Contenidos
- Discovering the Sum of Interior Angles in a Polygon
- Sum of the Interior Angles in a Quadrilateral
- Sum of the Interior Angles in a Pentagon
- Sum of the Interior Angles in a Hexagon
- Sum of the Interior Angles in a Heptagon
- Sum of the Interior Angles in an Octagon
- Moving on to Bigger Polygons
- Generalizing and Finding a Pattern
- Practical Practice
Sum of the Interior Angles in a Quadrilateral
Connect vertex A to each of the other vertices in the quadrilateral with a segment by moving the red sliders to the right
What did we just do?
By drawing segments that connect vertex A to all the other vertices, we have broken up this quadrilateral into triangles! Now connecting point A to point B and then point A to point C didn't really do anything, since they were already connected. That means we were only able to draw in one line that wasn't already there. We call segment AC a diagonal of the quadrilateral.
Think About and Answer the Following Questions
1. What do you remember about the sum of the angles in a triangle?
1. How many sides does a quadrilateral have?
3. How many triangles were you able to create in the quadrilateral above?
4. What might you guess then would be the sum of all the angles in a quadrilateral? (Use you answers from questions #1 and #3)
Generalizing and Finding a Pattern
Let's recap what we have discovered so far
[table][tr][td][u]Number of Sides of Polygon[/u][/td][td][u]Number of Triangles in Polygon[/u][/td][td][u]Sum of the Interior Angles[/u][/td][/tr][tr][td][center][b]4[/b][/center][/td][td][center][b][color=#ff00ff]2[/color][/b][/center][/td][td][center][color=#0000ff][b]180[/b][/color][math]^{\circ}\times2=360^{\circ}[/math][/center][/td][/tr][tr][td][center][b]5[/b][/center][/td][td][center][b][color=#ff00ff]3[/color][/b][/center][/td][td][center][b][color=#0000ff]180[/color][/b][math]^{\circ}\times3=540^{\circ}[/math][/center][/td][/tr][tr][td][center][b]6[/b][/center][/td][td][center][b][color=#ff00ff]4[/color][/b][/center][/td][td][center][b][color=#0000ff]180[/color][/b][math]^{\circ}\times4=720^{\circ}[/math][/center][/td][/tr][tr][td][center][b]7[/b][/center][/td][td][center][b][color=#ff00ff]5[/color][/b][/center][/td][td][center][b][color=#0000ff]180[/color][/b][math]^{\circ}\times5=900^{\circ}[/math][/center][/td][/tr][tr][td][center][b]8[/b][/center][/td][td][center][color=#ff00ff][b]6[/b][/color][/center][/td][td][center][b][color=#0000ff]180[/color][/b][math]^{\circ}\times6=1080^{\circ}[/math][/center][/td][/tr][tr][td][center][b]n[/b][/center][/td][td][center][b][color=#ff00ff]?[/color][/b][/center][/td][td][center][b][color=#0000ff]?[/color][/b][/center][/td][/tr][/table]
Do you notice a pattern?
How does the number of sides of a polygon compare to the number of triangles that can be drawn in that polygon? (How does the number in the first column compare to the number in the second column?)
If the number of sides of a polygon is [math]n[/math], then how many triangles can be drawn in that polygon? Write your answer in terms of [math]n[/math].
What number are we always multiplying the amount of triangles in a polygon by to get the sum of the interior angles in a polygon? (hint: what number always appears in the third column of the table?)
If the number of sides of a polygon is [math]n[/math], then what is the sum of the interior angles of that polygon? Write your answer in terms of [math]n[/math]. (Hint: use your answer from question 2)
And that's it!
You have just found the formula to find the sum of the interior angles in any polygon that has [math]n[/math] sides! Again, the formula for the sum of the interior angles of a polygon with [math]n[/math] sides is[br][br][center][math]180^\circ\times\left(n-2\right)[/math][/center]