Angles in Polygons

Count the triangular regions, not the diagonals.

A hexagon has six sides. How many triangles can you make from one vertex?

Use (number of sides − 2) × 180.

A quadrilateral contains two triangles.

Regular means the five interior angles are equal.

One exterior angle at every vertex adds up to a full turn.

1. The exterior angles total 360°.
2. A regular hexagon has 6 equal exterior angles, so each is 360\div6=60°.

Imagine walking around the outside and returning to your starting direction.

1. A complete walk around a convex polygon makes one full turn.
2. Its exterior angles total 360°.

Find how many equal 40-degree turns make 360 degrees.

1. The exterior angles total 360°.
2. The number of sides is 360\div40=9.

Find the adjacent exterior angle first.

1. An interior angle and its adjacent exterior angle total 180°.
2. The exterior angle is 180-150=30°.
3. The number of sides is 360\div30=12.

An angle sum does not tell you that the angles are equal.

1. The sum 540° applies to all pentagons.
2. Dividing by five gives the average angle. It gives every angle only when the angles are equal, as in a regular pentagon.

Find the pentagon total, then subtract all four known angles.

1. The pentagon total is (5-2)\times180=540°.
2. The known angles total 95+112+121+108=436°.
3. The missing angle is 540-436=104°.

Use either the interior sum or the 360-degree exterior sum.

Part (a)

1. The interior sum is (8-2)\times180=1080°.
2. Each angle is 1080\div8=135°.

Part (b)

1. An interior angle and its adjacent exterior angle total 180°.
2. Each exterior angle is 180-135=45°.

The hexagon is not stated to be regular, so subtract the known angles from the total.

1. The hexagon interior total is (6-2)\times180=720°.
2. The five given angles total 120+135+110+145+100=610°.
3. The sixth angle is 720-610=110°.

Find the angle contributed by each shape, then subtract from a full turn.

1. The equilateral triangle contributes 60° and the square contributes 90°.
2. The regular pentagon contributes (5-2)\times180\div5=108°.
3. Together these angles total 60+90+108=258°.
4. The gap is 360-258=102° (angles around a point).

Which total belongs to all the exterior angles around the polygon?

1. Exterior angles total 360°, not 180°.
2. The correct number of sides is 360\div30=12.

First work backwards to the number of sides. Then share the total equally.

1. The number of triangles is 2340\div180=13.
2. Since a convex polygon with n sides splits into n-2 triangles, it has 13+2=15 sides.
3. The polygon is regular, so each interior angle is 2340\div15=156°.

The interior and exterior angles together are six equal parts of 180 degrees.

1. Let the exterior angle be x°. The interior angle is 5x°.
2. x+5x=180, so 6x=180 and x=30.
3. The number of sides is 360\div30=12.

Subtract the square and octagon angles from 360 degrees, then work backwards.

1. The square angle is 90°. The regular octagon angle is 180-360\div8=135°.
2. The third interior angle is 360-90-135=135°.
3. Its exterior angle is 180-135=45°, so the polygon has 360\div45=8 sides.

Find the triangle angle at A before finding the polygon’s exterior angle.

1. The triangle angle at A is 180-90-36=54°, because angles in a triangle add to 180°.
2. The polygon interior angle at A is 90+54=144°.
3. Its exterior angle is 180-144=36°.
4. The number of sides is 360\div36=10.

Compare three pentagon angles and four pentagon angles with a full turn.

Part (a)

1. Each regular pentagon angle is (5-2)\times180\div5=108°.
2. The three angles total 3\times108=324°.
3. The gap is g=360-324=36°.

Part (b)

1. Three corners total 324°, leaving a gap. Four total 432°, causing overlap.
2. No whole number of these 108° corners totals 360°.