https://file.aiquickdraw.com/mathbot/file/1726651639790fziuelxu.jpg

The image shows the formula for the area of a triangle using Heron's formula, which is given as:

\[
\text{Area of a triangle} = \sqrt{s(s - a)(s - b)(s - c)}
\]

where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle, and \( s \) is the semi-perimeter:

\[
s = \frac{a + b + c}{2}
\]

You are given the sides of the triangle as \( a = 3 \), \( b = 7 \), and \( c = 8 \).

Let's calculate the semi-perimeter \( s \), and then use it to find the area of the triangle.

### Step 1: Calculate the semi-perimeter \( s \):
\[
s = \frac{3 + 7 + 8}{2} = \frac{18}{2} = 9
\]

### Step 2: Apply Heron's formula:
\[
\text{Area} = \sqrt{9(9 - 3)(9 - 7)(9 - 8)} = \sqrt{9 \times 6 \times 2 \times 1}
\]
\[
\text{Area} = \sqrt{9 \times 12} = \sqrt{108} \approx 10.39
\]

So, the area of the triangle is approximately **10.39 square units**.

Would you like further details or clarification on any part of this?

### Related Questions:
1. How does Heron’s formula work for a triangle with different side lengths?
2. What is the significance of the semi-perimeter \( s \) in Heron’s formula?
3. Can Heron’s formula be used for all triangles, including right-angled triangles?
4. What happens when the sum of two sides is less than the third side?
5. How can you determine if a given set of three sides forms a valid triangle?

**Tip:** Always check that the sum of any two sides of a triangle is greater than the third side to confirm that the sides form a valid triangle.

What is the area of a triangle with sides of 3, 7, and 8? Use Heron's formula and give your answer to two decimal places.

Learn how to calculate the area of a triangle with sides of 3, 7, and 8 using Heron's formula. This step-by-step guide includes key geometry concepts, such as semi-perimeter calculation and the application of Heron's Theorem, suitable for students in Grades 9-12.

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