Regular polygons, with their equal sides and equal angles, appear frequently in geometry and real-world applications. Understanding how to calculate their area is a fundamental skill. This comprehensive guide will provide a step-by-step approach to finding the area of any regular polygon, along with explanations, examples, and helpful tips. Whether you’re a student, a teacher, or simply someone looking to brush up on your geometry, this article will equip you with the knowledge and tools you need.
What is a Regular Polygon?
Before diving into the calculations, let’s define what a regular polygon is. A polygon is a closed, two-dimensional shape with straight sides. A regular polygon is a polygon where all sides are of equal length, and all interior angles are of equal measure. Examples of regular polygons include:
- Equilateral triangle (3 sides)
- Square (4 sides)
- Regular pentagon (5 sides)
- Regular hexagon (6 sides)
- Regular heptagon (7 sides)
- Regular octagon (8 sides)
- And so on…
Methods for Finding the Area of Regular Polygons
There are several methods for calculating the area of a regular polygon. We’ll focus on two primary methods:
- Using the Apothem
- Using Trigonometry
Method 1: Using the Apothem
The apothem is a line segment from the center of the polygon to the midpoint of one of its sides. It is always perpendicular to that side. Knowing the apothem and the perimeter of the polygon allows us to calculate the area efficiently.
Formula
The formula for the area (A) of a regular polygon using the apothem (a) and the perimeter (P) is:
A = (1/2) * a * P
Where:
- A = Area of the regular polygon
- a = Apothem (the distance from the center to the midpoint of a side)
- P = Perimeter (the total length of all the sides)
Steps to Calculate the Area Using the Apothem
- Find the Apothem (a): The apothem might be given in the problem, or you might need to calculate it. We’ll discuss how to calculate the apothem later in this article.
- Find the Perimeter (P): Determine the length of one side of the polygon (s) and multiply it by the number of sides (n) to find the perimeter. P = n * s
- Apply the Formula: Substitute the values of the apothem (a) and the perimeter (P) into the formula A = (1/2) * a * P.
- Calculate the Area (A): Perform the calculation to find the area of the regular polygon. Remember to include the appropriate units (e.g., square inches, square centimeters).
Example 1: Finding the Area of a Regular Pentagon
Let’s say we have a regular pentagon with a side length of 6 cm and an apothem of 4.1 cm. Find the area.
- Apothem (a): a = 4.1 cm (given)
- Perimeter (P): The pentagon has 5 sides, each 6 cm long. P = 5 * 6 cm = 30 cm
- Apply the Formula: A = (1/2) * a * P = (1/2) * 4.1 cm * 30 cm
- Calculate the Area (A): A = 61.5 cm²
Therefore, the area of the regular pentagon is 61.5 square centimeters.
Example 2: Finding the Area of a Regular Octagon
Consider a regular octagon with a side length of 8 inches and an apothem of 9.7 inches. Calculate the area.
- Apothem (a): a = 9.7 inches (given)
- Perimeter (P): The octagon has 8 sides, each 8 inches long. P = 8 * 8 inches = 64 inches
- Apply the Formula: A = (1/2) * a * P = (1/2) * 9.7 inches * 64 inches
- Calculate the Area (A): A = 310.4 inches²
Thus, the area of the regular octagon is 310.4 square inches.
Method 2: Using Trigonometry
If you don’t know the apothem directly, you can use trigonometry to find it, given the side length and the number of sides. This method relies on dividing the regular polygon into congruent isosceles triangles.
Understanding the Geometry
Imagine drawing lines from the center of the regular polygon to each of its vertices. This divides the polygon into ‘n’ congruent isosceles triangles, where ‘n’ is the number of sides. The apothem is the height of each of these isosceles triangles. The base of each triangle is one side of the polygon.
Finding the Central Angle
The angle at the center of the polygon (the vertex angle of each isosceles triangle) is called the central angle. The measure of the central angle is found by dividing 360 degrees by the number of sides (n):
Central Angle = 360° / n
Using Trigonometry to Find the Apothem
Now, consider one of the isosceles triangles. The apothem bisects the central angle and the base (side) of the triangle. This creates two right triangles. We can use trigonometry to find the length of the apothem. Let’s focus on one of these right triangles.
- The angle opposite the side that is half the side of the polygon is half of the central angle: (360° / n) / 2 = 180° / n
- The side opposite this angle is half the length of a side of the polygon: s / 2
- The apothem is the adjacent side to this angle.
We can use the tangent function to relate the angle, the opposite side, and the adjacent side (the apothem):
tan(angle) = opposite / adjacent
tan(180° / n) = (s / 2) / a
Solving for the apothem (a):
a = (s / 2) / tan(180° / n)
Formula for Area Using Trigonometry
Now that we have the apothem (a) in terms of the side length (s) and the number of sides (n), we can substitute this into the area formula A = (1/2) * a * P. Remember that P = n * s.
A = (1/2) * [(s / 2) / tan(180° / n)] * (n * s)
Simplifying, we get:
A = (n * s²) / (4 * tan(180° / n))
Where:
- A = Area of the regular polygon
- n = Number of sides
- s = Length of one side
Steps to Calculate the Area Using Trigonometry
- Determine the Number of Sides (n) and Side Length (s): Identify the number of sides of the regular polygon and the length of each side.
- Calculate 180° / n: Divide 180 degrees by the number of sides.
- Calculate tan(180° / n): Find the tangent of the angle calculated in the previous step. Make sure your calculator is in degree mode.
- Calculate s²: Square the side length.
- Apply the Formula: Substitute the values of n, s², and tan(180° / n) into the formula A = (n * s²) / (4 * tan(180° / n)).
- Calculate the Area (A): Perform the calculation to find the area of the regular polygon. Remember to include the appropriate units.
Example 3: Finding the Area of a Regular Hexagon
Let’s find the area of a regular hexagon with a side length of 5 inches.
- Number of Sides (n) and Side Length (s): n = 6, s = 5 inches
- Calculate 180° / n: 180° / 6 = 30°
- Calculate tan(180° / n): tan(30°) ≈ 0.5774
- Calculate s²: 5² = 25
- Apply the Formula: A = (n * s²) / (4 * tan(180° / n)) = (6 * 25) / (4 * 0.5774)
- Calculate the Area (A): A ≈ 64.95 inches²
Therefore, the area of the regular hexagon is approximately 64.95 square inches.
Example 4: Finding the Area of a Regular Decagon
Calculate the area of a regular decagon (10 sides) with a side length of 4 cm.
- Number of Sides (n) and Side Length (s): n = 10, s = 4 cm
- Calculate 180° / n: 180° / 10 = 18°
- Calculate tan(180° / n): tan(18°) ≈ 0.3249
- Calculate s²: 4² = 16
- Apply the Formula: A = (n * s²) / (4 * tan(180° / n)) = (10 * 16) / (4 * 0.3249)
- Calculate the Area (A): A ≈ 123.11 cm²
The area of the regular decagon is approximately 123.11 square centimeters.
Calculating the Apothem When Not Given
As we discussed, the apothem is crucial for calculating the area using the first method. If the apothem is not provided directly, here’s how to calculate it:
Using Trigonometry (Revisited)
This is essentially what we did in Method 2. If you know the side length (s) and the number of sides (n), you can use the formula we derived earlier:
a = (s / 2) / tan(180° / n)
Using the Pythagorean Theorem (If you know the radius)
If you know the radius (r) of the circumscribed circle (the circle that passes through all the vertices of the polygon) and the side length (s), you can also use the Pythagorean theorem. The radius, apothem, and half the side length form a right triangle.
a² + (s/2)² = r²
Solving for the apothem (a):
a = √(r² – (s/2)²)
Tips and Tricks
- Calculator Mode: Always ensure your calculator is in degree mode when using trigonometric functions.
- Units: Pay close attention to the units. If the side length is in centimeters, the area will be in square centimeters.
- Approximations: When using trigonometric functions, you’ll often get decimal approximations. Round your final answer to an appropriate number of decimal places.
- Special Cases: For squares, the apothem is simply half the side length. For equilateral triangles, you can use the formula for the area of an equilateral triangle (A = (√3 / 4) * s²) or the methods described above.
- Visualize: Drawing a diagram of the regular polygon can help you visualize the apothem, the triangles, and the relationships between the sides, angles, and apothem.
Real-World Applications
Calculating the area of regular polygons has practical applications in various fields, including:
- Architecture: Determining the amount of material needed to cover floors or walls with polygonal shapes.
- Engineering: Calculating the cross-sectional area of bolts, nuts, and other mechanical parts with polygonal shapes.
- Landscaping: Planning and estimating the amount of materials needed for polygonal gardens or paving stones.
- Manufacturing: Designing and fabricating objects with precise polygonal dimensions.
- Tiling: Calculating how many tiles of a certain regular polygon shape are needed to cover an area.
Common Mistakes to Avoid
- Forgetting Units: Always include the appropriate units (e.g., square inches, square meters) in your final answer.
- Incorrect Calculator Mode: Ensure your calculator is in degree mode when using trigonometric functions.
- Misidentifying Sides: Double-check that you are using the correct side length and number of sides for the polygon.
- Using the Wrong Formula: Choose the appropriate formula based on the information you have (apothem and perimeter, or side length and number of sides).
- Rounding Errors: Avoid rounding intermediate calculations too early, as this can lead to significant errors in the final answer.
Advanced Topics
For those interested in delving deeper into the subject, here are some advanced topics related to regular polygons:
- Inscribed and Circumscribed Circles: Understanding the relationship between regular polygons and the circles that can be inscribed within them or circumscribed around them.
- Tessellations: Exploring how regular polygons can be used to create tessellations (tilings) of a plane.
- Geometric Constructions: Learning how to construct regular polygons using compass and straightedge. Not all regular polygons can be constructed this way (e.g., a regular heptagon).
- Complex Numbers: Representing vertices of regular polygons as complex numbers and using complex number arithmetic to study their properties.
Conclusion
Calculating the area of regular polygons is a fundamental skill in geometry with numerous practical applications. By understanding the concepts of apothem, perimeter, central angles, and trigonometric functions, you can accurately determine the area of any regular polygon. Practice the examples provided and remember the tips and tricks to master this essential geometric skill. Whether you are calculating areas for construction, design, or simply satisfying your mathematical curiosity, the knowledge gained here will prove invaluable.