Sum of the Measures of the Exterior Angles of a Polygon
The knowledge of exterior angles may throw light on the nature of interior angles and sides.
A polygon say, a pentagon ABCDE is drawn on the floor, using a piece of chalk.
We stand at A and start walking along the side AB. On reaching B, we need to turn through an angle (equal to the exterior angle subtended by side BC wrt x-axis), to walk along BC.
When you reach at C, we need to turn through an angle (equal to the exterior angle subtended by side CD wrt y-axis) to walk along CD. We continue to move in this manner, until we return to side AB.
We would have in fact made one complete turn i.e.
Therefore,
The sum of the measures of the external angles of any polygon is 360°
This is true whatever be the number of sides of the polygon, as this same logic can be used for others as well.
Example 1: Find measure of x.
Solution: x + 90° + 50° + 110° =
x +
x =
Thus, x is equal to 110°.
Try These
1.What is the sum of the measures of its exterior angles x, y, z, p, q, r?
For the given regular hexagon,
The measure of ∠a =
The ∠x, ∠y, ∠z, ∠p, ∠q, ∠r in the given figure, are
x ,y ,z ,p ,q ,r are all exterior angles.
Hence, x + y + z + p + q + r =
2. Is x = y = z = p = q = r? Why?
It is a hexagon with all sides
a + r = a + x = a + y = a + z = a + p = a + q =
Hence, x = y = z = p = q = r (after
3. What is the measure of each?
(i) exterior angle
(ii) interior angle
Solution:
(i) The sum of the exterior angles of any polygon is always
For a regular polygon, all exterior angles are equal.
Therefore, the measure of each exterior angle is given by:
Exterior Angle =
For a hexagon, the number of sides is
Thus,
Exterior Angle =
(ii) The sum of the interior angles of a polygon with n sides is given by:
Sum of Interior Angles = (n−2) × 180°
For a hexagon, n = 6.
Sum of Interior Angles = (6−2) × 180° =
Since the hexagon is regular, all interior angles are equal.
Interior Angle =
Thus, for a regular hexagon: the measure of each exterior angle is 60° while measure of each interior angle is 120°.
4. Repeat this activity for the cases of:
(i) a regular octagon
(ii) a regular 20-gon
Solution:
(i)
Exterior Angle:
Exterior Angle =
Interior Angle:
Sum of Interior Angles = (n−2) × 180° = (8−2) × 180° =
Interior Angle =
Thus, for a regular octagon: the measure of each exterior angle is 45° while measure of each interior angle is 135°.
(ii)
Exterior Angle:
Exterior Angle =
Interior Angle:
Sum of Interior Angles = (n−2) × 180° = (20−2) × 180° =
Interior Angle =
Thus, for a regular 20-gon: the measure of each exterior angle is 18° while measure of each interior angle is 162°.
Example 2: Find the number of sides of a regular polygon whose each exterior angle has a measure of 45°.
Solution: Total measure of all exterior angles =
Measure of each exterior angle =
Therefore, the number of exterior angles =
The polygon has 8 sides.