Question content area top
Part 1
A triangular building is bounded by three streets. The building measures approximately 93 feet on the first​ street, 185 feet on the second​ street, and 178 feet on the third street. Approximate the ground area A covered by the building.
Round to the nearest hundredth

To approximate the area \( A \) of a triangular building with sides of known lengths, we can use **Heron's formula**, which states:

\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]

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

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

For this problem:
- \( a = 93 \) feet,
- \( b = 185 \) feet,
- \( c = 178 \) feet.

### Step 1: Calculate the Semi-perimeter \( s \)
\[
s = \frac{93 + 185 + 178}{2} = \frac{456}{2} = 228
\]

### Step 2: Apply Heron’s Formula
Substitute \( s = 228 \), \( a = 93 \), \( b = 185 \), and \( c = 178 \) into the formula:

\[
A = \sqrt{228 \times (228 - 93) \times (228 - 185) \times (228 - 178)}
\]

Breaking it down:
\[
A = \sqrt{228 \times 135 \times 43 \times 50}
\]

Calculating inside the square root:
1. \( 228 \times 135 = 30780 \)
2. \( 30780 \times 43 = 1323540 \)
3. \( 1323540 \times 50 = 66177000 \)

\[
A = \sqrt{66177000} \approx 8135.15
\]

### Answer
The approximate ground area covered by the building is **8135.15 square feet**.

Would you like more details on any of the steps, or have further questions?

### Related Questions:
1. How is Heron’s formula derived?
2. What are alternative methods to calculate the area of a triangle?
3. How would the area change if the sides of the triangle were slightly different?
4. How does the unit of measurement affect calculations in Heron’s formula?
5. Could this formula apply to non-triangular polygons?

### Tip:
In problems involving triangles with known side lengths, Heron’s formula is a quick way to find the area without needing the height.

A triangular building is bounded by three streets. The building measures approximately 93 feet on the first street, 185 feet on the second street, and 178 feet on the third street. Approximate the ground area A covered by the building. Round to the nearest hundredth.

Find the ground area of a triangular building with side lengths of 93 feet, 185 feet, and 178 feet using Heron’s formula. This solution breaks down each step, including semi-perimeter calculation and area approximation, making it easy for high school students to follow. Perfect for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation