Geometry, the study of shapes and spaces, is a language of lines, angles, and forms that structures our world. From the organic patterns of a snowflake to the intentional design of a soccer ball, polygons are everywhere. Among these, the pentagon—a five-sided polygon—holds a unique fascination. It is a symbol of strength and biology, found in the Pentagon building and the cross-section of an okra pod. But beyond its symbolic presence, understanding the pentagon requires delving into its mathematical heart: its angles, area, and the calculations that define it.
This article will demystify the regular pentagon, explore its key properties, and provide the tools to perform essential calculations.
What is a Pentagon?
A pentagon is any two-dimensional closed shape with five straight sides and five angles. However, the most studied and symmetric type is the regular pentagon. Its defining characteristics are:
- All five sides are of equal length.
- All five interior angles are of equal measure (108° each).
- It can be inscribed inside a circle (cyclic).
The sum of the interior angles of any pentagon, regular or irregular, is always 540°. This is derived from the formula for polygon interior angles: (n-2) × 180°, where n is the number of sides. For n=5: (5-2) × 180° = 3 × 180° = 540°.
In a regular pentagon, since all angles are equal, each interior angle is 540° ÷ 5 = 108°.
Key Calculations for a Regular Pentagon
Let’s define the key variables we’ll use in our formulas:
- s = length of one side
- a = apothem (the distance from the center to the midpoint of any side, perpendicular to that side)
- P = Perimeter
- A = Area
- R = Radius of the circumscribed circle (circumradius)
- r = Radius of the inscribed circle (inradius, which is the same as the apothem,
a)
1. Perimeter (P)
This is the simplest calculation. The perimeter is the sum of the lengths of all sides.
P = 5 × s
2. Area (A)
Calculating the area is slightly more complex. The standard formula for the area of any regular polygon is:
Area = (½) × Perimeter × Apothem or A = (½) × P × a
Therefore, for a pentagon:
A = (½) × 5s × a = (5/2) × s × a
But how do we find the apothem a? This requires trigonometry. By dividing the pentagon into five identical isosceles triangles from the center, we can focus on one triangle. The central angle is 72° (360°/5). The apothem splits this triangle and the side s in half, creating a right triangle.
- Half of the side = s/2
- The angle at the center is half of 72°, which is 36°.
- In this right triangle: tan(36°) = (opposite/adjacent) = (s/2) / a
Solving for the apothem a:
a = (s/2) / tan(36°) = s / (2 × tan(36°))
Plugging this back into the area formula gives us a formula that depends only on the side length s:
A = (5/2) × s × [s / (2 × tan(36°))] = (5 × s²) / (4 × tan(36°))
Using an approximate value for tan(36°) ≈ 0.7265, the formula simplifies to:
A ≈ 1.72048 × s²
3. Finding the Apothem (a) and Circumradius (R)
As shown above, the apothem can be found if the side length is known:
a = s / (2 × tan(36°)) ≈ s / 1.453 ≈ 0.688 × s
The circumradius R (the distance from the center to a vertex) can also be found using trigonometry. In the same right triangle:
sin(36°) = (opposite/hypotenuse) = (s/2) / R
Solving for R:
R = (s/2) / sin(36°) = s / (2 × sin(36°))
Using an approximate value for sin(36°) ≈ 0.5878:
R ≈ s / 1.1756 ≈ 0.851 × s
Practical Calculation Example
Problem: A regular pentagon has a side length (s) of 10 cm. Find its perimeter, apothem, and area.
1. Perimeter:
P = 5 × s = 5 × 10 cm = 50 cm
2. Apothem:
a = s / (2 × tan(36°)) = 10 / (2 × 0.7265) ≈ 10 / 1.453 ≈ 6.88 cm
3. Area (using the apothem):
A = (½) × P × a = (½) × 50 cm × 6.88 cm ≈ 172.05 cm²
Area (using the side-length formula):
A ≈ 1.72048 × s² = 1.72048 × (10)² = 1.72048 × 100 ≈ 172.05 cm²
Both methods yield the same result, confirming our calculations.
(FAQs)
1. What is the difference between a pentagon and a regular pentagon?
All regular pentagons are pentagons, but not all pentagons are regular. A “pentagon” is the general term for any five-sided polygon. Its sides and angles can be of any length and measure, as long as it’s closed and has five sides. A “regular pentagon” is a specific type of pentagon where all sides and all angles are identical, making it perfectly symmetrical.
2. Why is the sum of interior angles always 540°?
This is a rule for all simple polygons (ones that don’t intersect themselves). You