Radius and Apothem of Regular Polygons
On this blog, let’s calculate the lengths of a regular polygon’s radius, apothem, and area knowing only the side length and internal angle.
A regular polygon is a polygon in which every side has an equal length, and every internal angle is equal.
Let x be the length of one side of the regular polygon, and θ be the internal angle of the polygon where:
θ = (n – 2) / n × 180°
The radius (r) of the regular polygon is a line segment from a vertex to the center of the polygon. The radius bisects the vertex, therefore cutting the internal angle in half.
The apothem (a) is a line segment from the center of the polygon to the center of the polygon’s line segment. If we extend the apothem beyond the border, the apothem splits the length of the side segment in half.
Zooming in, a right triangle is formed between the radius, apothem, and half of the polygon line segment.
By trigonometry:
tan (θ / 2) = a / (x / 2)
a = (x / 2) × tan (θ / 2)
and
cos (θ / 2) = (x / 2) ÷ r
r = x / (2 × cos (θ / 2))
Knowing the apothem, the area of the regular polygon is:
area = perimeter × a / 2
where the perimeter = n × x
Then:
area = (n × x) × a / 2
= 1 / 2 × n × x × a
= 1 / 2 × n × x × (x / 2 × tan(θ / 2))
= 1 / 4 × n × x^2 × tan(θ / 2)
Another Formula for an Area’s Regular Polygons
The area of a regular polygon is often stated as:
area = 1 / 4 × n × x^2 / (tan (180° / n)) = 1 / 4 × n × x^2 × cot (180° / n))
We can show that the two formulas for area are equivalent:
Note that:
θ = (n – 2) / n × 180°
θ = 180° - 360° / n
Divide both sides by 2:
θ / 2 = 90° - 180° / n
180° / n = 90° - θ / 2 [ I ]
Observe that the trigonometric identity, for any angle α:
tan(90° - α) = 1 / tan(α) = cot(α)
and
cot(90° - α) = 1 / cot(α) = tan(α) [ II ]
Then:
area = 1 / 4 × n × x^2 × tan(θ / 2)
= 1 / 4 × n × x^2 × cot(90° - θ / 2) [ II ]
= 1 / 4 × n × x^2 × cot(180° / n) [ I ]
= 1 / 4 × n × x^2 / tan(180° / n) [ I ]
In Summary:
Internal Angle: θ = (n – 2) / 2 × 180°
Apothem: a = (x / 2) × tan (θ / 2)
Radius: r = x / (2 × cos (θ / 2))
Area = 1 / 4 × n × x^2 × tan(θ / 2) = 1 / 4 × n × x^2 / tan(180° / n)
(Note: 180° = π radians)
Table of Apothem and Radius, with side length of 1
n |
θ (in degrees) |
θ / 2 (in degrees) |
a (to 5 decimal places) |
r (to 5 decimal places) |
3 |
60 |
30 |
0.28868 |
0.57735 |
4 |
90 |
45 |
0.5 |
0.70711 |
6 |
120 |
60 |
0.86603 |
1 |
8 |
135 |
67.5 |
1.20711 |
1.30656 |
12 |
150 |
75 |
1.86603 |
1.93185 |
Eddie
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