Understanding the Concepts of Regular Polygons

The Beauty of Pentagons

Pentagons are fascinating polygons with five sides and five vertices. The shape and classification of pentagons can vary, with some being convex and others concave. Convex pentagons have interior angles that are all less than 180°, while concave pentagons have at least one angle greater than 180°. When all sides and interior angles of a pentagon are equal, it is considered regular. However, if a pentagon has equal sides but is concave, it is called equilateral. Any pentagon that does not fall under these classifications is irregular.

Types of pentagons

The sum of the internal angles of a pentagon, regardless of its classification, is always constant at 540°. This can be proved by dividing the pentagon into three non-overlapping triangles. No matter the arrangement of these triangles, their internal angles will always add up to 540°.

A pentagon can be divided into three triangles

Exploring the Properties of Regular Pentagons

Symmetry: An Axis of Beauty

Regular pentagons possess a unique symmetry. They have five axes of symmetry that pass through a vertex and the midpoint of the opposite edge. These axes intersect at the center of the pentagon, which is also its center of gravity or centroid.

Axes of symmetry of regular pentagon

Interior and Central Angles: A Perfect Pair

In regular pentagons, all interior angles are equal. The sum of these interior angles is always 540°. Therefore, each interior angle, denoted as φ, measures 108°. Furthermore, by drawing lines from the center of the pentagon to each vertex, we create five identical triangles. The central angle, denoted as θ, of each triangle is 72°.

In each triangle, the remaining two angles are identical, measuring 54°. The sum of all angles in the triangle is 180° (72° + 54° + 54°), which is also half of the interior angle φ (108°/2 = 54°). Importantly, the sum of the interior angle φ and the central angle θ is always 180°. In other words, φ and θ are supplementary angles.

Interior and central angle of a regular pentagon

Circumcircle and Incircle: Encircling the Beauty

A regular pentagon possesses two significant circles. The circumcircle, or the circumscribed circle, passes through all five vertices of the pentagon. Its center coincides with the center of the pentagon, where all axes of symmetry intersect. On the other hand, the inscribed circle, or the incircle, touches tangentially to all five edges of the pentagon at their midpoints. The centers of both circles are the same, lying at the center of the pentagon. The radius of the circumcircle is called the circumradius, while the radius of the incircle is the inradius.

Circumcircle and incircle of a regular pentagon

The relationships between the side length (a) of a regular pentagon and its circumradius (Rc) and inradius (Ri) can be determined. Using basic trigonometry, the following expressions can be derived for any regular polygon:

• Rc = a / (2 sin(θ/2))
• Ri = a / (2 tan(θ/2))
• Ri = Rc cos(θ/2)

For the regular pentagon specifically, with θ = 72°, the approximations for these expressions are:

• Rc ≈ 0.851a
• Ri ≈ 0.688a
• Ri ≈ 0.809Rc

Area and Perimeter: Measuring the Beauty

To find the area of a regular pentagon, we divide it into five identical isosceles triangles. Each triangle has one side (a) and two sides equivalent to the circumradius (Rc). The height of each triangle, cast from the vertex lying at the pentagon center, is equal to the inradius (Ri). Therefore, the area of each triangle is (1/2) a Ri. The total area of the five triangles is:

A = (5/2) a Ri A ≈ 1.720 * a^2

The perimeter of any N-sided regular polygon is simply the sum of the lengths of all sides: P = N * a. For the regular pentagon, the perimeter is 5a.

Area and perimeter of a regular pentagon

The Bounding Box: The Perfect Fit

The bounding box of a regular pentagon is the smallest rectangle that completely encloses the shape. While the dimensions of the bounding box can be intuitively drawn, finding the exact measurements requires some calculations.

Height: Reaching New Heights

The height (h) of a regular pentagon is the distance from one vertex to the opposite edge. It is perpendicular to the opposite edge and passes through the center of the pentagon. By definition, the distance from the center to a vertex is the circumradius (Rc), and the distance from the center to an edge is the inradius (Ri). Therefore, the expression for the height is:

h = Rc + Ri

The height can be expressed in terms of the circumradius (Rc), inradius (Ri), or side length (a) using the respective analytical expressions. The following formulas can be derived:

• h = Rc * (1 + cos(θ/2))
• h = Ri * (1 + (1/cos(θ/2)))
• h = (a/2) * (1 + cos(θ/2)) / sin(θ/2)

Approximating the value of θ to 72°, we obtain the following approximations:

• h ≈ 1.809 * Rc
• h ≈ 2.236 * Ri
• h ≈ 1.539 * a

Width: Expanding Horizons

The width (w) of a regular pentagon is the distance between two opposite vertices (the length of its diagonal). To find this distance, we can use a right triangle formed by extending one side of the pentagon. The hypotenuse of this right triangle is the side length (a) of the pentagon. By considering the adjacent interior angle φ, which is supplementary to the central angle θ, we can calculate the length (w1) of the triangle side:

w1 = a * cos(θ)

Finally, the total width (w) can be obtained by adding twice the length (w1) to the side length (a), as the triangle to the right of the pentagon is identical to the one examined: w = a + 2 a cos(θ)

Approximating θ to 72°, we get the following approximation: w ≈ 1.618 * a

Drawing a Regular Pentagon: Unleash Your Creativity

Drawing a regular pentagon with a given side length (a) is achievable using simple drawing tools. Follow these steps:

1. Draw a line segment with a length equal to the desired side length (a).
2. Extend the line segment to the left.
3. Construct a circular arc with its center at the right end of the line segment and a radius equal to the segment's length.
4. Repeat the previous step, but this time with the center point at the left end of the line segment.
5. Draw a line perpendicular to the line segment, passing through the intersection of the two arcs. The line should cross the line segment at its midpoint.
6. Draw another line perpendicular to the line segment, passing through the left end of the line segment. Mark the intersection point with the circular arc drawn in step 4.
7. Draw a new circular arc by placing one needle of the compass at the midpoint of the line segment (found in step 5) and the drawing tip at the intersection marked in step 6. Rotate the compass until it intersects with the extended line segment drawn in step 2. Mark this new intersection.
8. Draw another circular arc by placing one needle of the compass at the right end of the line segment and the drawing tip at the intersection marked in step 7. Rotate the compass clockwise, marking two intersections: one with the arc drawn in step 4, and another with the line drawn in step 5. These two intersections represent two vertices of the pentagon.
9. Using the second intersection as the compass needle and the first one as the drawing tip (both intersections marked in the previous step), draw a circular arc that intersects with the arc drawn in step 3. Mark this new intersection as another vertex of the pentagon.
10. The two ends of the line segment, along with the three intersections marked in steps 8 and 9, represent the five vertices of the regular pentagon. Connect them with straight lines to complete the shape.

Drawing a regular pentagon given its side length (a)

Please note that the described procedure is not strictly a "ruler and compass" construction. Steps 5 and 6 involve the use of a triangle to draw perpendicular lines. However, if a strict geometric drawing by "ruler and compass" is required, the use of a triangle can be replaced with the method for constructing perpendicular lines using a ruler and compass alone.

The Beauty of Regular Polygons: A Quick Reference

Here is a concise list of the main formulas and helpful approximations related to regular pentagons:

• Circumradius (Rc) ≈ 0.851a
• Inradius (Ri) ≈ 0.688a
• Height (h) ≈ 1.539a
• Width (w) ≈ 1.618a
• Area (A) ≈ 1.720a^2

Refer to this quick reference to explore the many possibilities and properties of regular pentagons.