|
|
Okay, so I need to figure out how to solve this problem where I have to find the area of a regular hexagon with side length 6 cm. Hmm, let's start by recalling what a regular hexagon is. A regular hexagon is a six-sided polygon where all sides are equal in length and all interior angles are equal too. Since it's regular, it should also be convex, meaning all the vertices point outward without any indentations.
Now, the area of a regular polygon can be calculated if I know the formula. I remember that for a regular polygon with 'n' sides of length 's', the area can be found using the formula: (n * s²) / (4 * tan(π/n)). Let me verify that. Yeah, that seems right because the formula accounts for dividing the polygon into 'n' isosceles triangles, each with a central angle of 2π/n radians. Then, the area of each triangle would be (1/2)s(s/(2*tan(π/n))), and multiplying by 'n' gives the total area. So, simplifying that, it becomes (n * s²) / (4 * tan(π/n)). Okay, that makes sense.
Alternatively, I might have seen another method where you divide the hexagon into six equilateral triangles. Wait, is that right? Let me think. If you connect the center of the hexagon to all its vertices, you divide it into six congruent equilateral triangles. Each triangle would have a side length equal to the side length of the hexagon, which is 6 cm here. If that's the case, then the area of the hexagon would be six times the area of one equilateral triangle.
Let me check both methods to see if they give the same result. Let's start with the first formula: (n * s²) / (4 * tan(π/n)). For a hexagon, n=6, s=6 cm. Plugging in, we get (6 * 6²) / (4 * tan(π/6)). Let's compute that step by step.
First, compute the numerator: 6 * 36 = 216.
Then, the denominator: 4 * tan(π/6). I know that tan(π/6) is tan(30°), which is 1/√3. So, 4 * (1/√3) = 4/√3.
Therefore, the area is 216 / (4/√3) = 216 * (√3/4) = (216/4) * √3 = 54√3 cm².
Now, using the second method: six equilateral triangles. The area of an equilateral triangle with side length 'a' is (√3/4) * a². So, for a=6 cm, the area is (√3/4) * 36 = 9√3 cm² per triangle. Multiply by six: 6 * 9√3 = 54√3 cm². Same result! Good, both methods agree. That gives me confidence that the answer is correct.
Wait, but why does the first method work? Let me make sure I understand the derivation. The formula (n * s²) / (4 * tan(π/n)) is derived by considering that each of the 'n' isosceles triangles has a base of length 's' and a central angle of 2π/n. The two equal sides of each triangle are the radii of the circumscribed circle around the polygon. To find the area of each triangle, we can split it into two right triangles by drawing an altitude from the center to the base. This altitude is the apothem of the polygon.
The apothem 'a' (which is the height of each of those isosceles triangles) can be calculated as (s)/(2 * tan(π/n)). Therefore, the area of each isosceles triangle is (1/2)baseheight = (1/2)s(s/(2tan(π/n))) = s²/(4tan(π/n)). Multiply by 'n' triangles: ns²/(4tan(π/n)). That's exactly the formula. So, that checks out.
Alternatively, when you split the hexagon into six equilateral triangles, it's because in a regular hexagon, the radius of the circumscribed circle (distance from center to vertex) is equal to the side length. That's a property specific to regular hexagons. So, each triangle formed by two adjacent vertices and the center is equilateral. Hence, each has area (√3/4)*s², and six of them give 54√3. That's another way to see it.
So, both methods are valid and lead to the same answer. Therefore, the area of the regular hexagon with side length 6 cm is 54√3 cm².
But just to make sure, let me compute it numerically. √3 is approximately 1.732. So, 54 * 1.732 ≈ 54 * 1.732. Let's compute that: 501.732=86.6, 41.732=6.928, so total ≈86.6+6.928≈93.528 cm². If I use another approach, say, using the apothem.
Wait, another formula for the area of a regular polygon is (perimeter * apothem)/2. Let's try that. The perimeter of the hexagon is 6*6=36 cm. The apothem is the distance from the center to the midpoint of a side, which is also the height of each equilateral triangle. In an equilateral triangle with side length 6 cm, the height is (√3/2)*s = (√3/2)*6 = 3√3 cm. So, the apothem is 3√3 cm.
Therefore, area = (36 * 3√3)/2 = (108√3)/2 = 54√3 cm². Same result again. So, three different methods all leading to 54√3. That's solid.
I think that's thorough. The key takeaway is recognizing that a regular hexagon can be divided into six equilateral triangles, which simplifies the calculation. Alternatively, using the general formula for any regular polygon also works. Either way, the answer is 54√3 cm².
The area of a regular hexagon with side length 6 cm can be calculated using multiple methods, all yielding the same result:
Using the formula for a regular polygon:
[
\text{Area} = \frac{n \cdot s^2}{4 \cdot \tan(\pi/n)}
]
For ( n = 6 ) and ( s = 6 ) cm:
[
\text{Area} = \frac{6 \cdot 6^2}{4 \cdot \tan(\pi/6)} = \frac{216}{4 \cdot (1/\sqrt{3})} = 54\sqrt{3} , \text{cm}^2
]
Dividing into six equilateral triangles:
Each triangle has area:
[
\frac{\sqrt{3}}{4} \cdot 6^2 = 9\sqrt{3} , \text{cm}^2
]
Total area:
[
6 \cdot 9\sqrt{3} = 54\sqrt{3} , \text{cm}^2
]
Using perimeter and apothem:
Perimeter ( = 6 \cdot 6 = 36 ) cm. Apothem ( = 3\sqrt{3} ) cm.
[
\text{Area} = \frac{1}{2} \cdot \text{Perimeter} \cdot \text{Apothem} = \frac{1}{2} \cdot 36 \cdot 3\sqrt{3} = 54\sqrt{3} , \text{cm}^2
]
Final Answer: The area of the regular hexagon is (\boxed{54\sqrt{3}}) cm².
|
|
发表于 2025-11-15 16:14:28