Suppose you are a chef preparing a special dish for a cooking competition. The dish is presented on a triangular-shaped plate with side lengths measuring
21
21​ inches,
17
17​ inches, and
28
28​ inches. Calculate the area of the triangular plate, rounding to the nearest tenth of a square inch

To calculate the area of a triangular-shaped plate with side lengths of 21 inches, 17 inches, and 28 inches, we can use **Heron's formula**.

### Steps:

1. **Find the semi-perimeter (s):**
   \[
   s = \frac{a + b + c}{2}
   \]
   where \(a = 21\), \(b = 17\), and \(c = 28\).

2. **Apply Heron's formula:**
   \[
   \text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
   \]

Let's calculate the area.

1. **Semi-perimeter (s):**
   \[
   s = \frac{21 + 17 + 28}{2} = \frac{66}{2} = 33
   \]

2. **Using Heron's formula:**
   \[
   \text{Area} = \sqrt{33(33 - 21)(33 - 17)(33 - 28)}
   \]
   \[
   = \sqrt{33(12)(16)(5)}
   \]
   \[
   = \sqrt{33 \times 12 \times 16 \times 5}
   \]
   \[
   = \sqrt{31680}
   \]

Now, calculating the square root:

\[
\text{Area} \approx \sqrt{31680} \approx 178.0 \text{ square inches}
\]

### Final Answer:
The area of the triangular plate is approximately **178.0 square inches**.

Would you like more details on this calculation or have any questions?

Here are 5 related questions to explore further:

1. How is Heron's formula derived from basic geometry?
2. Can you apply Heron’s formula to any triangle? What are the conditions?
3. What other methods can be used to calculate the area of a triangle?
4. How does the accuracy of different triangle area formulas compare?
5. What is the significance of semi-perimeter in triangle area calculation?

**Tip:** Always check if the given side lengths satisfy the triangle inequality theorem before proceeding with calculations!

Suppose you are a chef preparing a special dish for a cooking competition. The dish is presented on a triangular-shaped plate with side lengths measuring 21 inches, 17 inches, and 28 inches. Calculate the area of the triangular plate, rounding to the nearest tenth of a square inch.

Learn how to calculate the area of a triangle with side lengths of 21 inches, 17 inches, and 28 inches using Heron's formula. This step-by-step solution covers key geometry concepts, including the calculation of the semi-perimeter and applying Heron's theorem. Suitable for students in Grades 9-12.

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