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Toolbox · Polygon angle calculator

Polygon angle calculator

Side count, interior sum or one interior angle as input; get interior sum, interior angle, exterior angle and diagonal count, with a drawn polygon

Formula

\Sigma=(n-2)\times180^\circ

Interior sum = (n - 2) x 180 degrees

Enter the side count n (n >= 3).

Presets

Tap an edge, angle or region to see how it is computed

Sides and interior sum

Interior and exterior angles

Diagonals

Relationα120°

\alpha=180^\circ-\frac{360^\circ}{6}=120^\circ

Interior sum720

Each interior angle120

Each exterior angle60

Diagonals9

n6Σ720α120β60D9

Steps

6 / 6

1Given
$n=6$
Collect the given quantities first.
2Angle sum
$\Sigma=(n-2)\times180^\circ=(6-2)\times180^\circ=720^\circ$
Triangle angles sum to 180 degrees; an n-gon sums to (n-2) x 180.
3Find angle
$\alpha=\frac{\Sigma}{n}=\frac{720}{6}=120^\circ$
Use a trig ratio or the cosine rule to get the angle.
4Angle sum
$\beta=180^\circ-\alpha=60^\circ=\frac{360^\circ}{6}$
Triangle angles sum to 180 degrees; an n-gon sums to (n-2) x 180.
5Diagonals
$D=\frac{n(n-3)}{2}=\frac{6\times3}{2}=9$
Use the diagonal formula or Pythagoras.
6Result
$n=6,\ \Sigma=720^\circ,\ \alpha=120^\circ,\ \beta=60^\circ,\ D=9$
Collect all computed quantities.

Concept

An n-gon has interior sum (n - 2) x 180 degrees, while the exterior sum is always 360 degrees regardless of n.

A regular polygon has equal sides and equal angles; each exterior angle is 360/n degrees.

Going further

Why n - 2: From one vertex you can draw n - 3 diagonals, cutting the polygon into n - 2 triangles of 180 degrees each.
Exterior sum is always 360: Walking around the polygon turns you through one full revolution, so exterior angles always add to 360 degrees.
Counting diagonals: Each vertex links to n - 3 others, giving n(n - 3) segments, but each diagonal is counted twice, so n(n - 3)/2.

Common mistakes

n is the number of sides, not vertices plus one.
Exterior and interior angles are supplementary.
Only regular polygons have equal interior angles; do not divide the sum by n otherwise.

Page keywordsinterior angle sumregular polygonpolygon anglesnumber of diagonalsinterior angle sum formulainterior angle of a dodecagonfind sides from angle sumwhy exterior angles sum to 360how many diagonals

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