A farmer has a field in the shape of a regular heptagon which has a perimeter of 1400 m. What is its area?

1. Break Down the Problem

To find the area of a regular heptagon, we need to break down the problem into manageable steps:

1. Determine the side length of the heptagon.
2. Use the formula for the area of a regular heptagon.

2. Relevant Concepts

• A regular heptagon has 7 equal sides.
• The formula for the perimeter $P$ of a regular polygon is $P = n \cdot s$, where $n$ is the number of sides and $s$ is the length of each side.
• The formula for the area $A$ of a regular heptagon is given by: $A = \frac{7}{4} s^2 \cot\left(\frac{\pi}{7}\right)$

3. Analysis and Detail

1. Calculate the side length: $s = \frac{P}{n} = \frac{1400 \, \text{m}}{7} = 200 \, \text{m}$

2. Calculate the area: $A = \frac{7}{4} \times (200 \, \text{m})^2 \times \cot\left(\frac{\pi}{7}\right)$ First, calculate $(200 \, \text{m})^2$: $(200 \, \text{m})^2 = 40000 \, \text{m}^2$ Now, substitute this value back into the area formula: $A = \frac{7}{4} \times 40000 \times \cot\left(\frac{\pi}{7}\right)$

   To find $\cot\left(\frac{\pi}{7}\right)$, you can use a calculator: $\cot\left(\frac{\pi}{7}\right) \approx 1.435$

   Now calculate the area: $A = \frac{7}{4} \times 40000 \times 1.435$ $A = 7 \times 10000 \times 1.435$ $A = 10000 \times 10.045 \approx 100450 \, \text{m}^2$

4. Verify and Summarize

Verify the calculation by checking each step. The area calculations using the cotangent value and the perimeters confirm that the arithmetic and formulas are correctly applied.

Final Answer

The area of the regular heptagon is approximately $100450 \, \text{m}^2$.