How to Calculate the Apothem of a Regular Hexagon
A strong grasp of the apothem helps students tackle questions on regular polygons and area calculation in school and competitive exams. Understanding this geometric line makes it easier to solve problems, especially when formulas require both the side and the center-to-edge distance for polygons.
Apothem Meaning and Definition
In geometry, the apothem is the perpendicular distance from the center of a regular polygon to the midpoint of one of its sides. Unlike the radius, which reaches the polygon’s vertex, the apothem always meets the side at a 90-degree angle. This concept is used not just for clarity in figures, but it’s essential in calculating the area of regular polygons, such as pentagons, hexagons, and more. For more on regular polygons, check out this detailed guide.
Formula Used in Apothem
The standard formula is: \( a = \frac{s}{2\, \tan\left(\frac{180}{n}\right)} \), where \( a \) is the apothem, \( s \) is the side length, and \( n \) is the number of sides. The apothem is also related to area by \( A = \frac{1}{2}aP \), where \( P \) is perimeter. These formulas are crucial for exam calculations. See area problems at area of polygon.
Here’s a helpful table to understand apothem more clearly:
Apothem Table
| Polygon Type | Formula for Apothem | Example with Side Length 6 |
|---|---|---|
| Square | \( \dfrac{s}{2} \) | 3 |
| Hexagon | \( \dfrac{s}{2\, \tan(30^\circ)} \) | \( \approx 5.20 \) |
| Pentagon | \( \dfrac{s}{2\, \tan(36^\circ)} \) | \( \approx 4.13 \) |
This table shows how the pattern of apothem calculation changes with different polygons, helping you compare formulas quickly. For more, visit polygon properties.
How to Find the Apothem in a Regular Polygon
To find an apothem step-by-step, follow this process (using a regular hexagon as an example):
1. Identify the side length and count the number of sides (for a hexagon, \( n = 6 \), \( s = 8 \)).
2. Use the apothem formula: \( a = \frac{s}{2\, \tan\left(\frac{180}{n}\right)} \).
3. Calculate the angle: \( \tan\left(\frac{180}{6}\right) = \tan(30^\circ) \).
4. Substitute the side length: \( a = \frac{8}{2\times 0.577} \approx \frac{8}{1.154} \approx 6.93 \).
5. The apothem is **approximately 6.93 units**.
Apothem vs. Radius
| Aspect | Apothem | Radius |
|---|---|---|
| Definition | Center to midpoint of side (perpendicular) | Center to vertex (corner) |
| Used For | Area calculations | Drawing polygons, circles |
| Always shorter? | Yes (for regular polygons) | No (always reaches the vertex) |
This comparison helps you distinguish between these two lines in polygons. Read more about polygon radii at perimeter of polygon.
Worked Example – Solving a Problem
Find the apothem of a regular pentagon with a side length of 10 cm.
1. Note side length: \( s = 10 \) and for a pentagon, \( n = 5 \).
2. Apply the formula: \( a = \frac{10}{2 \tan(36^\circ)} \).
3. Calculate \( \tan(36^\circ) \approx 0.7265 \).
4. Compute denominator: \( 2 \times 0.7265 = 1.453 \).
5. Divide: \( \frac{10}{1.453} \approx 6.88 \).
Final Answer: The apothem is 6.88 cm.
Practice Problems
- Calculate the apothem of a regular hexagon where each side is 12 cm.
- Use the apothem to find the area of a square whose side is 8 cm.
- Compare the radius and apothem of a regular triangle with side length 6 cm.
- Find the perimeter of a pentagon if its apothem is 7 cm and side length is 10 cm.
Common Mistakes to Avoid
- Confusing apothem with the radius. The radius goes to the vertex, while the apothem is always perpendicular to a side.
- Plugging in the number of sides incorrectly in the formula.
- Forgetting to convert angle measures to degrees if your calculator is set to radians.
Real-World Applications
The concept of apothem is practical in architecture, engineering, and tiling. It helps calculate materials for floor tiles (which are polygons), as well as in design and construction. Vedantu's curated approaches show where maths like this matters outside your textbook.
We explored the idea of apothem, its key role in geometry, and step-by-step ways to use it in polygon area and perimeter problems. With support from Vedantu, you can deepen your understanding and be ready for any polygon-based exam question.
For more on related topics, see:
- Regular Polygon
- Area of Polygon
- Perimeter of Polygon
- Polygons
- Interior Angles of a Polygon
FAQs on What Is an Apothem in Geometry?
1. What is an apothem in geometry?
Apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any of its sides. It is used in calculating the area of regular polygons like pentagons, hexagons, and squares.
2. How do you find the apothem of a regular polygon?
To find the apothem of a regular polygon, use the formula: Apothem = side length / (2 × tan(π/n)), where n is the number of sides. For polygons like square, pentagon, or hexagon, substitute the respective values of side and n.
3. Is the apothem equal to the radius of a polygon?
No, apothem is generally not equal to the radius of a polygon (which is the distance from the center to a vertex). However, in a square or when the number of sides is very large, the apothem and radius values become nearly equal, but for polygons like pentagons and hexagons, they are different.
4. What does an apothem look like?
In a regular polygon, the apothem is a line drawn from the polygon’s center perpendicular to the midpoint of any of its sides. It always forms a right angle with the side and divides the side into two equal parts.
5. What is the apothem of a hexagon?
The apothem of a regular hexagon (with side length a) is calculated using the formula: Apothem = a × (√3 / 2). It helps in finding the area and other properties of a hexagon.
6. What is the formula for the apothem of a pentagon?
For a regular pentagon with side length a, the apothem is given by: Apothem = a / (2 × tan(π/5)). Substitute the side value to calculate the apothem.
7. How is the apothem of a square calculated?
For a square with side length a, the apothem is simply a/2, as it is the distance from the center to the midpoint of any side, also known as the distance to the side.
8. What role does the apothem play in finding the area of a regular polygon?
The area of a regular polygon can be found using its apothem with the formula: Area = 0.5 × Perimeter × Apothem. Here, the perimeter is the total length of all sides of the polygon.
9. Is there an apothem in a triangle?
For an equilateral triangle, the apothem is the perpendicular drawn from the center to one side, calculated by side / (2 × tan(π/3)) or equivalently side × (√3 / 6). For non-equilateral triangles, apothem is not defined.
10. How do apothem and radius differ in regular polygons?
In a regular polygon, the radius refers to the distance from the center to any vertex, while the apothem is from the center to the midpoint of a side. Radius is always longer than the apothem, except in a square, where both may be equal.
11. Is there an apothem calculator formula for regular polygons?
Yes, use the formula: Apothem = side / (2 × tan(π/n)), where 'side' is the length of one polygon side and 'n' is the total number of sides. Many online apothem calculators use this equation for quick calculation.
12. Why is the concept of apothem important in geometry?
The apothem helps in deriving the area formula for regular polygons and simplifies solving complex geometry problems involving polygons. It also relates to the radius, side, and angles, forming the basis for advanced geometric applications.
