Bob wants to build an ice skating rink in his backyard, but his wife says he can only use the part beyond the wood-chipped path running through their yard. What would be the area of his rink if it is triangular-shaped with sides of length 21
feet, 24
feet, and 27
feet? Round to the nearest square foot.

To find the area of Bob's triangular-shaped ice skating rink with sides of lengths 21 feet, 24 feet, and 27 feet, we can use Heron's formula. 

Heron's formula states that the area of a triangle can be calculated from its side lengths \(a\), \(b\), and \(c\) using the following steps:

1. Calculate the semi-perimeter \(s\):
   \[
   s = \frac{a + b + c}{2}
   \]
2. Use the semi-perimeter to find the area \(A\):
   \[
   A = \sqrt{s(s-a)(s-b)(s-c)}
   \]

Let's apply these steps:

1. Calculate the semi-perimeter \(s\):
   \[
   s = \frac{21 + 24 + 27}{2} = \frac{72}{2} = 36
   \]

2. Calculate the area \(A\):
   \[
   A = \sqrt{36(36-21)(36-24)(36-27)}
   \]
   \[
   A = \sqrt{36 \times 15 \times 12 \times 9}
   \]

To simplify this calculation:
\[
36 \times 15 = 540
\]
\[
540 \times 12 = 6480
\]
\[
6480 \times 9 = 58320
\]

Thus, 
\[
A = \sqrt{58320}
\]

Approximating the square root:
\[
A \approx 241.43
\]

Rounding to the nearest square foot:
\[
A \approx 241 \text{ square feet}
\]

So, the area of Bob's ice skating rink is approximately 241 square feet.

Would you like more details or have any questions?

Here are 8 related questions that expand on this information:
1. How does Heron's formula work for different types of triangles?
2. Can you calculate the height of the triangle from one of its sides?
3. What is the significance of the semi-perimeter in Heron's formula?
4. How would the area change if one of the side lengths was increased by 5 feet?
5. Can Heron's formula be used for right triangles?
6. What other methods can be used to find the area of a triangle?
7. How would you find the circumradius of this triangle?
8. How can you verify if the given side lengths form a valid triangle?

**Tip:** When using Heron's formula, ensure all calculations are accurate, especially when working with large numbers or decimals.

Learn how to calculate the area of a triangular ice skating rink with sides of lengths 21 feet, 24 feet, and 27 feet using Heron's formula. Get a detailed step-by-step solution and understand the application of Heron's formula in solving triangle area problems. Suitable for Grades 10-12 students and those interested in geometric formulas.

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