At a Glance - Angles in a Polygon
As we discussed before, the three angles of a triangle always add up to 180°.
In each case . By the way, means "the measurement of angle A".
To find the total number of degrees in any polygon, all we have to do is divide the shape into triangles. To do this start from any vertex and draw diagonals to all non-adjacent vertices.
|Here is a quadrilateral.|
|If we draw all the diagonals from a vertex we get two triangles.|
|Each triangle has 180°, so 2 ×180° = 360° in a quadrilateral.|
|Pentagon – 5 sides||3 triangles × 180° = 540°|
|Hexagon – 6 sides||4 triangles × 180° = 720°|
|Septagon – 7 sides||5 triangles × 180° = 900°|
|Octagon – 8 sides||6 triangles × 180° = 1080°|
Are you noticing a pattern? Turns out, the number of triangles formed by drawing the diagonals is two less than the number of sides. If we use the variable n to equal the number of sides, then we could find a formula to calculate the number of degrees in any polygon:
What is the sum of the angles in a dodecagon?
A dodecagon has 12 sides, so .
What is the measure of each angle in a regular nonagon?
A nonagon has 9 sides. Using our formula, .
Find the missing angle.
A triangle has 180°. If we add the measures of angles I and J and subtract from 180, we get:
Find the measurement of angle Q
This is a hexagon. The total number of degrees equals:
Find the missing angles in this isosceles trapezoid.
Since this is an isosceles trapezoid, angles I and L are congruent. In addition, angles J and K are congruent, so