In this activity, you will experiment with finding the sum of the measures of the interior angles of different polygons. You can use this sum to find missing angle measure in polygons.

  1. In the Gizmotm, make sure Number of sides is set to 3. (To quickly set a value, type a number in the box to the right of the slider and press ENTER.) Unselect Regular Polygon. Drag A, B, and C to form any triangle that you like.
    1. How many angles does the triangle have? Measure each of the angles by clicking on Click to measure angles and using the Gizmo protractors. (For help using the protractor, click on Gizmo Help, below the Gizmo.) Record your data. Click on Show angle measures and, with Interior angles selected, check your answers.
    2. What is the sum of the angle measures of this triangle? Drag A, B, and C to make a variety of new triangles. Do the angle measures for all triangles add up to 180°?
    3. On paper, make a table with three columns. Label the columns as shown below. Fill in the triangle information you found. At this point, your table should look like this:

      table has 3 columns: figure; number of sides; and sum of the measures of the interior angles.  according to the table, a triangle has 3 sides and the sum of the measures of the interior angles of a triangle is 180 degrees.  a quadrilateral has 4 sides.  The sum of the measures of the interior angles of a quadrilateral is left blank.  a pentagon has 5 sides.  the sum of the measures of the interior angles of a pentagon is left blank.  a hexagon has 6 sides.  the sum of the measure of the angles of a hexagon
  2. Select Regular polygon. With Show angle measures and Interior angles selected, increase the Number of sides to 4.
    1. A 4-sided polygon is called a quadrilateral. A polygon with all sides congruent and all angles congruent is called a regular polygon. What is another name for a regular quadrilateral?
    2. What are the measures of the angles of a square? What is the sum of all four interior angle measures of a square?
    3. Turn off Regular polygon. Drag the vertices of the quadrilateral to make one that you like. Find the sum of the measures of the interior angles. What is that sum? Is this the same sum as a square has?
    4. Enter the sum in the correct place in the table.
  3. Increase the Number of sides to 5. Unselect Show angle measures.
    1. A quicker way to find the sum of the measures of the interior angles of a polygon is to calculate (n − 2) •180°, where n is the number of sides. Use the formula to find the sum of the interior angle measures of a pentagon. What value will you use in place of n? Calculate the sum and enter that sum in the table.
    2. Use the formula to finish filling in the table for the other polygons listed there. Click on Show angle sum table to check the accuracy of your table. To have the Gizmo complete the table for you, select Show angle measures and, with Interior angles selected, move the Number of sides slider back and forth.
  4. Use what you have learned about the sum of the interior angle measures of polygons to solve these problems.
    1. If the angle measures of the interior angles of an octagon are 122°, 143°, 152°, 107°, 128°, 130°, and 145°, find the missing angle measure. Explain your method.
    2. Find the measure of each angle in a regular hexagon. To check your answer, set Number of sides to 6, click on Regular polygon, and select Show angle measures.
    3. What is the measure of each interior angle in a regular pentagon? Use the Gizmo to check your answer.
  5. Why do you think the formula for the sum of the interior angle measures of a polygon is (n − 2) • 180°? If you can see the reason, it will help you remember the formula. To investigate, be sure that Regular polygon is selected, and turn on Divide into triangles. Also, turn off Show angle measures. Then slowly move the Number of sides slider back and forth.
    1. How many triangles can a quadrilateral be divided into? A pentagon? A hexagon? How do these answers compare to the number of sides in each polygon?
    2. What is the sum of the interior angle measures of a triangle?
    3. See if you can put this together into an explanation, in words, of why the sum of the interior angle measures of a polygon equals (n − 2) • 180°.