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You are building a flower bed. It is going to be triangular.



You want to know how much topsoil to get, so you need to figure out the area of the triangle.



You measure all three sides and find them to be: a = 5ft, b = 7ft, and c = 8ft.



Draw out a triangle and label it.

What is the area of the triangle? (make sure to include units).



To solve for the area of the triangle, we can use **Heron's Formula**. This formula calculates the area of a triangle when all three sides are known.

### Step-by-Step Solution:

1. **Label the Triangle**  
   First, let's draw and label the triangle with sides \( a = 5 \) feet, \( b = 7 \) feet, and \( c = 8 \) feet.

   ```
          c = 8 ft
       A___________C
       |          /
       |        /
     a=5ft    /
       |    /
       |  /
       |/
      B
       b=7ft
   ```

2. **Calculate the Semi-Perimeter \( s \)**  
   The semi-perimeter \( s \) is half of the triangle's perimeter:
   \[
   s = \frac{a + b + c}{2}
   \]
   Substituting the values:
   \[
   s = \frac{5 + 7 + 8}{2} = \frac{20}{2} = 10 \text{ ft}
   \]

3. **Apply Heron's Formula**  
   Heron's formula for the area \( A \) of a triangle is:
   \[
   A = \sqrt{s(s - a)(s - b)(s - c)}
   \]
   Substituting the values of \( s \), \( a \), \( b \), and \( c \):
   \[
   A = \sqrt{10(10 - 5)(10 - 7)(10 - 8)}
   \]
   Simplify inside the parentheses:
   \[
   A = \sqrt{10 \times 5 \times 3 \times 2}
   \]
   Continue calculating:
   \[
   A = \sqrt{10 \times 5 \times 3 \times 2} = \sqrt{300}
   \]
   Taking the square root:
   \[
   A \approx 17.32 \text{ square feet}
   \]

### Final Answer:
The area of the triangle is approximately **17.32 square feet**.

---

Would you like additional details on any of these steps?

Here are five related questions to explore further:
1. How would the area change if side lengths were different?
2. What other methods can we use to find the area of a triangle?
3. How can we use trigonometry to verify the area of this triangle?
4. What is the importance of knowing the triangle's semi-perimeter?
5. How would you calculate the perimeter of the triangle?

**Tip:** Heron’s formula is most useful when you know all three sides but no angles of a triangle.

You are building a flower bed. It is going to be triangular. You want to know how much topsoil to get, so you need to figure out the area of the triangle. You measure all three sides and find them to be: a = 5ft, b = 7ft, and c = 8ft. Draw out a triangle and label it. What is the area of the triangle? (make sure to include units).

Learn how to calculate the area of a triangular flower bed with sides measuring 5 feet, 7 feet, and 8 feet using Heron's formula. This geometry problem demonstrates step-by-step how to find the area when all sides are known, suitable for students in Grades 8-10.

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