Finding the Area of a Triangular Field

The problem asks us to find the area of a triangular field when the lengths of its three sides are given. The side lengths are 15m, 19m, and 22m. To find the area of a triangle when all three sides are known, we can use Heron's formula.

Understanding Heron's Formula

Heron's formula is a useful tool for calculating the area of a triangle when only the lengths of the three sides are provided. The formula is given by:

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

Where:

• \( A \) is the area of the triangle.
• \( a, b, \) and \( c \) are the lengths of the three sides of the triangle.
• \( s \) is the semi-perimeter of the triangle.

The semi-perimeter \( s \) is half the perimeter of the triangle. It is calculated as:

\( s = \frac{a+b+c}{2} \)

Step-by-Step Calculation

Let the side lengths of the triangular field be \( a = 15 \) m, \( b = 19 \) m, and \( c = 22 \) m.

Step 1: Calculate the Semi-Perimeter (s)

First, we find the semi-perimeter \( s \):

\( s = \frac{15 + 19 + 22}{2} = \frac{56}{2} = 28 \) m

Step 2: Calculate the Terms (s-a), (s-b), and (s-c)

Next, we calculate the differences between the semi-perimeter and each side length:

• \( s - a = 28 - 15 = 13 \) m
• \( s - b = 28 - 19 = 9 \) m
• \( s - c = 28 - 22 = 6 \) m

Step 3: Apply Heron's Formula

Now, we substitute these values into Heron's formula:

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

\( A = \sqrt{28 \times 13 \times 9 \times 6} \)

Let's calculate the product inside the square root:

\( 28 \times 13 = 364 \)

\( 9 \times 6 = 54 \)

\( A = \sqrt{364 \times 54} \)

\( 364 \times 54 = 19656 \)

\( A = \sqrt{19656} \)

Step 4: Calculate the Square Root and Round

Calculating the square root of 19656:

\( \sqrt{19656} \approx 140.199857... \)

The question asks for the area to be corrected to one decimal place. Rounding 140.199857... to one decimal place gives 140.2 m².

Thus, the area of the triangular field is approximately 140.2 m².

Revision Table: Triangular Field Area Calculation

Additional Information: Area of Triangles

There are several ways to calculate the area of a triangle, depending on the information available:

• Base and Height: If the base (b) and corresponding height (h) are known, the area is \( A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} bh \). This is the most common formula.
• Two Sides and Included Angle: If two sides (a, b) and the angle (\( \gamma \)) between them are known, the area is \( A = \frac{1}{2} ab \sin(\gamma) \). Similarly for other pairs of sides and their included angles.
• Three Sides (Heron's Formula): As used in this problem, if all three sides (a, b, c) are known, the area can be calculated using Heron's formula: \( A = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = (a+b+c)/2 \). This formula is particularly useful when the height is not easily determined.

Heron's formula is powerful because it only requires side lengths, which are often simpler to measure in real-world scenarios like surveying a field.