C4.6; E4.6 - Angles in polygons

C4.6; E4.6 - Angles in polygons

A regular polygon's angles follow from its number of sides, so you can calculate them without measuring. You will build interior-angle sums from triangles, understand exterior angles as turns, and work backwards from an angle to the number of sides.

Interior angle sums

Could a regular polygon have an interior angle of 160∘160^\circ? The answer will reveal exactly how many sides it needs. Start with the total amount of angle inside a polygon.

An interior angle lies inside the polygon at a vertex. Choose one vertex of a convex polygon and draw diagonals to every non-neighbouring vertex. The diagonals stay inside and split the polygon into triangles without gaps or overlaps. A pentagon makes three triangles; adding one side makes room for one more triangle.

[DIAGRAM: asset_name: Triangulating convex polygons; asset_slug: polygon_triangle_fans; caption: A fan from one vertex contains two fewer triangles than the number of sides.; description: A regular hexagon, octagon and decagon are each divided from one chosen vertex into respectively four, six and eight non-overlapping triangles. Diagonals are dashed blue and each diagram labels its triangle count.]
A fan from one vertex contains two fewer triangles than the number of sides.

Interior-angle sum

For a convex polygon with nn sides, the fan contains n−2n-2 triangles, so

S=(n−2)×180∘.S=(n-2)\times180^\circ.

Here nn is a whole number at least 33. The formula gives a total, not an individual angle.

Not given in the List of formulas: understand the triangle derivation.

Why subtract two? Of the other n−1n-1 vertices, the two neighbours connect by sides already present. The remaining n−3n-3 diagonals divide the interior into one more region than the number of diagonals: n−2n-2 triangles. We use regular convex polygons here; regular means equal sides and equal angles.

Worked example 1: the sum for a hexagon

Question. Calculate the sum of the interior angles of a regular hexagon.

Plan. Count the triangles from one vertex, then multiply by their angle sum.

triangles=6−2=4fan from one vertexS=4×180∘=720∘each triangle contributes 180∘\begin{aligned} \text{triangles}&=6-2=4 && \text{fan from one vertex}\\ S&=4\times180^\circ=720^\circ && \text{each triangle contributes }180^\circ \end{aligned}

Answer. 720∘720^\circ.

Check. Dividing equally between six angles gives 120∘120^\circ each. Six such angles total 720∘720^\circ.

Change only the side count. Each additional side contributes one more triangle to the fan.

Worked example 2: the sum for an octagon

Question. Calculate the sum of the interior angles of a regular octagon.

Plan. Replace six sides by eight in the triangle count.

triangles=8−2=6two more than the hexagonS=6×180∘=1080∘sum the six triangles\begin{aligned} \text{triangles}&=8-2=6 && \text{two more than the hexagon}\\ S&=6\times180^\circ=1080^\circ && \text{sum the six triangles} \end{aligned}

Answer. 1080∘1080^\circ.

Check. There are two extra sides compared with the hexagon, so the total increases by 2×180∘=360∘2\times180^\circ=360^\circ. Indeed, 720+360=1080720+360=1080.

Watch out

“An octagon's angle is 1080∘1080^\circ” confuses a total with one angle. The total is 1080∘1080^\circ; regularity is needed before dividing it equally among eight vertices.

Keep the triangle count separate from the side count.

Exterior angles and a full turn

Walk round a convex polygon, continuing along each side and turning at the corners. When you arrive back facing your original direction, your turns total one full turn, 360∘360^\circ. At each vertex, the turning angle is the exterior angle between a side's forward extension and the next side.

Interior and exterior angles

At a vertex of a convex polygon, an interior angle ii and its adjacent exterior angle ee lie on a straight line:

i+e=180∘.i+e=180^\circ.

One exterior turning angle at each vertex, taken consistently round the polygon, totals 360∘360^\circ.

Not given in the List of formulas: connect the straight angle with the full turn.

[DIAGRAM: asset_name: Interior angles and outside turns; asset_slug: regular_exterior_turns; caption: Extend the incoming side forwards to identify the adjacent exterior angle.; description: A regular pentagon, hexagon and octagon each have one incoming side extended forwards at a highlighted vertex. Interior arcs are labelled 108 degrees, 120 degrees and 135 degrees, while adjacent exterior arcs are labelled e. Small arrows show consistent anticlockwise travel around each polygon.]
Extend the incoming side forwards to identify the adjacent exterior angle.

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