Finding the Perimeter of a Square Garden Using Triangle Area

The problem asks us to find the perimeter of a square-shaped garden whose area is equal to the area of a triangular park with given side lengths. First, we need to calculate the area of the triangular park. Since we are given the lengths of all three sides of the triangle, we can use Heron's formula.

Calculating the Area of the Triangular Park

The sides of the triangular park are given as a = 60 m, b = 297 m, and c = 303 m.

Heron's formula for the area of a triangle with sides a, b, and c is:

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

where s is the semi-perimeter of the triangle, calculated as:

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

Let's calculate the semi-perimeter (s):

\(s = \frac{60 + 297 + 303}{2} = \frac{660}{2} = 330 \text{ m}\)

Now, let's calculate the terms inside the square root:

• \(s - a = 330 - 60 = 270\)
• \(s - b = 330 - 297 = 33\)
• \(s - c = 330 - 303 = 27\)

Now, we can calculate the area using Heron's formula:

\(\text{Area}_{\text{triangle}} = \sqrt{330 \times 270 \times 33 \times 27}\)

Let's simplify the expression inside the square root:

\(\text{Area}_{\text{triangle}} = \sqrt{(33 \times 10) \times (27 \times 10) \times 33 \times 27}\)

\(\text{Area}_{\text{triangle}} = \sqrt{33^2 \times 27^2 \times 10^2}\)

\(\text{Area}_{\text{triangle}} = \sqrt{(33 \times 27 \times 10)^2}\)

\(\text{Area}_{\text{triangle}} = 33 \times 27 \times 10\)

\(\text{Area}_{\text{triangle}} = 891 \times 10\)

\(\text{Area}_{\text{triangle}} = 8910 \text{ m}^2\)

Calculating the Perimeter of the Square Garden

The problem states that the area of the square-shaped garden is equal to the area of the triangular park.

\(\text{Area}_{\text{square}} = \text{Area}_{\text{triangle}} = 8910 \text{ m}^2\)

Let the side length of the square garden be 'a'. The area of a square is given by \(a^2\).

\(a^2 = 8910\)

To find the side length 'a', we take the square root of the area:

\(a = \sqrt{8910}\)

Let's simplify the square root:

\(8910 = 81 \times 110\)

\(a = \sqrt{81 \times 110}\)

\(a = \sqrt{81} \times \sqrt{110}\)

\(a = 9 \times \sqrt{110}\)

\(a = 9\sqrt{110} \text{ m}\)

The perimeter of a square is given by \(4 \times \text{side length}\).

\(\text{Perimeter}_{\text{square}} = 4 \times a\)

\(\text{Perimeter}_{\text{square}} = 4 \times (9\sqrt{110})\)

\(\text{Perimeter}_{\text{square}} = 36\sqrt{110} \text{ m}\)

Thus, the perimeter of the square-shaped garden is \(36\sqrt{110}\) m.

Revision Table: Area and Perimeter Calculations

Additional Information: Heron's Formula and Square Properties

Heron's Formula: This formula is particularly useful for finding the area of a triangle when only the lengths of its three sides are known. It avoids the need to calculate angles or the altitude of the triangle. The formula is named after Hero of Alexandria.

Properties of a Square: A square is a quadrilateral with four equal sides and four right (\(90^\circ\)) angles. All sides are parallel to opposite sides. The area of a square is the side length squared (\(a^2\)), and the perimeter is four times the side length (\(4a\)).

This problem connects the concepts of area calculation for different geometric shapes, specifically using Heron's formula for a triangle and the basic area and perimeter formulas for a square.