HEPTADECAGON: PROPERTIES, DIAGONALS, PERIMETER, AREA - MATHS

The heptadecagon is a regular polygon with 17 sides and 17 vertices. Its construction can be done in the Euclidean style, that is, using only the ruler and the compass. It was the great mathematical genius Carl Friedrich Gauss (1777-1855), barely 18 years old, who found the procedure for its construction in 1796.

Apparently, Gauss was always very inclined to this geometric figure, to such an extent that from the day he discovered its construction he decided to be a mathematician. It is also said that he wanted the heptadecagon to be engraved on his tombstone.

Figure 1. The heptadecagon is a regular polygon with 17 sides and 17 vertices. Source: F. Zapata.

Gauss also found the formula to determine which regular polygons have the possibility of being constructed with ruler and compass, since some do not have exact Euclidean construction.

Characteristics of the heptadecagon

As for its characteristics, like any polygon, the sum of its internal angles is important. In a regular polygon with n sides, the sum is given by:

This sum, expressed in radians, looks like this:

From the above formulas it can be easily deduced that each internal angle of a heptadecagon has an exact measure α given by:

It follows that the internal angle roughly is:

Diagonals and perimeter

Diagonals and perimeter are other important aspects. In any polygon the number of diagonals is:

D = n (n - 3) / 2 and in the case of the heptadecagon, as n = 17, we then have that D = 119 diagonals.

On the other hand, if the length of each side of the heptadecagon is known, then the perimeter of the regular heptadecagon is found simply by adding 17 times that length, or what is equivalent to 17 times the length d of each side:

P = 17 d

Perimeter of the heptadecagon

Sometimes only the radius r of the heptadecagon is known, so it is necessary to develop a formula for this case.

To this end, the concept of apothem is introduced. The apothem is the segment that goes from the center of the regular polygon to the midpoint of one side. The apothem relative to one side is perpendicular to that side (see figure 2).

Figure 2. The parts of a regular polygon with radius r and its apothem are shown. (Own elaboration)

In addition, the apothem is the bisector of the angle with central vertex and sides on two consecutive vertices of the polygon, this allows finding a relationship between the radius r and the side d.

If the central angle DOE is called β and taking into account that the apothem OJ is a bisector, we have EJ = d / 2 = r Sen (β / 2), from which we have a relationship to find the length d of the side of a polygon known its radius r and its central angle β:

d = 2 r Sen (β / 2)

In the case of the heptadecagon β = 360º / 17, we have:

d = 2 r Sen (180º / 17) ≈ 0.3675 r

Finally, the formula for the perimeter of the heptadecagon is obtained, known its radius:

P = 34 r Sen (180º / 17) ≈ 6.2475 r

The perimeter of a heptadecagon is close to the perimeter of the circumference that circumscribes it, but its value is less, that is, the perimeter of the circumscribed circle is Pcir = 2π r ≈ 6.2832 r.

Area

To determine the area of the heptadecagon we will refer to Figure 2, which shows the sides and apothem of a regular polygon with n sides. In this figure the triangle EOD has an area equal to the base d (side of the polygon) times the height a (apothem of the polygon) divided by 2:

EOD area = (dxa) / 2

So, knowing the apothem a of the heptadecagon and the side d of the same, its area is:

Heptadecagon area = (17/2) (dxa)

Area given the side

To get a formula for the area of the heptadecagon knowing the length of its seventeen sides, it is necessary to obtain a relationship between the length of the apothem a and the side d.

With reference to figure 2, the following trigonometric relationship is obtained:

Tan (β / 2) = EJ / OJ = (d / 2) / a, where β is the central angle DOE. So the apothem a can be calculated if the length d of the side of the polygon and the central angle β are known:

a = (d / 2) Cotan (β / 2)

If this expression is now substituted for the apothem, in the formula for the area of the heptadecagon obtained in the previous section, we have:

Heptadecagon area = (17/4) (d ²) Cotan (β / 2)

Being β = 360º / 17 for the heptadecagon, so we finally have the desired formula:

Heptadecagon area = (17/4) (d ²) Cotan (180º / 17)

Area given the radius

In the previous sections a relationship had been found between the side d of a regular polygon and its radius r, this relationship being the following:

d = 2 r Sen (β / 2)

This expression for d is inserted in the expression obtained in the previous section for the area. If the relevant substitutions and simplifications are made, the formula that allows calculating the area of the heptadecagon is obtained:

Heptadecagon area = (17/2) (r ²) Sen (β) = (17/2) (r ²) Sen (360º / 17)

An approximate expression for the area is:

Heptadecagon area = 3.0706 (r ²)

As expected, this area is slightly less than the area of the circle circumscribing the heptadecagon A _circ = π r ² ≈ 3.1416 r ². To be precise, it is 2% less than that of its circumscribed circle.

Examples

Example 1

To answer the question it is necessary to remember the relationship between the side and the radius of a regular n-sided polygon:

d = 2 r Sen (180º / n)

For the heptadecagon n = 17, so d = 0.3675 r, that is, the radius of the heptadecagon is r = 2 cm / 0.3675 = 5.4423 cm or

10.8844 cm in diameter.

The perimeter of a 2-cm side heptadecagon is P = 17 * 2 cm = 34 cm.

Example 2

We must refer to the formula shown in the previous section, which allows us to find the area of a heptadecagon when it has the length d of its side:

Heptadecagon area = (17/4) (d ²) / Tan (180º / 17)

By substituting d = 2 cm in the previous formula, we obtain:

Area = 90.94 cm

References

CEA (2003). Geometry elements: with exercises and compass geometry. University of Medellin.
Campos, F., Cerecedo, FJ (2014). Mathematics 2. Grupo Editorial Patria.
Freed, K. (2007). Discover Polygons. Benchmark Education Company.
Hendrik, V. (2013). Generalized Polygons. Birkhäuser.
IGER. (sf). Mathematics First Semester Tacaná. IGER.
Jr. geometry. (2014). Polygons. Lulu Press, Inc.
Miller, Heeren, & Hornsby. (2006). Mathematics: Reasoning And Applications (Tenth Edition). Pearson Education.
Patiño, M. (2006). Mathematics 5. Editorial Progreso.
Sada, M. 17-sided regular polygon with ruler and compass. Recovered from: geogebra.org
Wikipedia. Heptadecagon. Recovered from: es.wikipedia.com

HEPTADECAGON: PROPERTIES, DIAGONALS, PERIMETER, AREA - MATHS - 2025