Alliance Form 3 P1 Q5 — Angles of a Regular Polygon

KCSE (Form 3) Form 3 Geometry

Published

The Question

“A regular polygon has an interior angle of 3x degrees and an exterior angle of (x - 20) degrees. Find the number of sides of the polygon.”

1

Relate interior and exterior angles

At any corner of a polygon, the interior angle and the exterior angle lie side by side on a straight line. If you extend one side of the polygon straight past the corner, those two angles together make a straight angle. Angles on a straight line always add up to 180 degrees, so the interior angle plus the exterior angle equals 180 degrees. This single fact lets us build an equation in x.

3x+(x−20)=1803x + (x - 20) = 180
2

Solve for x

Combine the two x-terms on the left, then move the constant across. Adding 20 to both sides isolates the term in x, and dividing by four gives the value of x. This value is what unlocks every other angle in the question.

4x−20=1804x - 20 = 180
4x=2004x = 200
x=50x = 50
3

Find the exterior angle

Now substitute the value of x back into the expression for the exterior angle. We use the exterior angle rather than the interior one because the exterior angles of a polygon obey a very simple rule that makes the last step easy.

x−20=50−20=30∘x - 20 = 50 - 20 = 30^\circ
4

Use the exterior-angle sum to count the sides

The exterior angles of any polygon always add up to 360 degrees, no matter how many sides it has. In a regular polygon every exterior angle is equal, so the number of sides is simply 360 divided by one exterior angle. Dividing 360 by 30 gives the number of sides.

n=360∘30∘=12n = \frac{360^\circ}{30^\circ} = 12

Final Result

The regular polygon has 12 sides (a regular dodecagon).

Why this method works

The method works because two independent facts pin the polygon down. First, an interior angle and its neighbouring exterior angle sit on a straight line, so they must total 180 degrees — this converts the unknown x into a real number. Second, the exterior angles of every polygon sweep once around a full turn, so they must total 360 degrees; because the polygon is regular, all those exterior angles are equal, and dividing the full turn by one of them counts how many corners (and therefore sides) there are.

✓

Interior angle is 3x = 150 degrees; 150 + 30 = 180 (angles on a line), and 12 x 30 = 360 (sum of exterior angles). Both checks hold.