Mensuration: Area and Perimeter - Apply Heron formula to find triangle area from side lengths

Find the area of a triangle whose sides are $13\text{ cm}$, $14\text{ cm}$, and $15\text{ cm}$.

1. Calculate semi-perimeter $s$: $s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\text{ cm}$
2. Apply Heron's Formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ $\text{Area} = \sqrt{21(21-13)(21-14)(21-15)}$ $\text{Area} = \sqrt{21 \times 8 \times 7 \times 6}$
3. Factorize to simplify: $\text{Area} = \sqrt{(3 \times 7) \times (2 \times 2 \times 2) \times 7 \times (2 \times 3)}$ Grouping pairs: $\text{Area} = \sqrt{7^2 \times 3^2 \times 2^4}$ $\text{Area} = 7 \times 3 \times 4 = 84$ Result: $84\text{ cm}^2$

We first identify the three sides $a, b, c$. We calculate the semi-perimeter $s$, then find the differences between the semi-perimeter and each side. By substituting these into Heron's formula and using prime factorization, we find the area without needing the height.

An isosceles triangle has a perimeter of $32\text{ cm}$ and the ratio of its equal side to its base is $3:2$. Find the area of the triangle.

1. Find side lengths: Let the sides be $3x, 3x,$ and $2x$. $3x + 3x + 2x = 32 \implies 8x = 32 \implies x = 4$ Sides are $12\text{ cm}, 12\text{ cm},$ and $8\text{ cm}$.
2. Find semi-perimeter $s$: $s = \frac{32}{2} = 16\text{ cm}$
3. Apply Heron's Formula: $\text{Area} = \sqrt{16(16-12)(16-12)(16-8)}$ $\text{Area} = \sqrt{16 \times 4 \times 4 \times 8}$
4. Simplify: $\text{Area} = \sqrt{16 \times 16 \times 4 \times 2} = 16 \times 2\sqrt{2} = 32\sqrt{2}$ Result: $32\sqrt{2}\text{ cm}^2$

We use the given ratio and total perimeter to solve for the individual side lengths. After finding the sides, we calculate the semi-perimeter and apply Heron's formula, simplifying the final radical for the result.

1. Identify sides: $a = 120\text{ m}$, $b = 80\text{ m}$, $c = 50\text{ m}$.
2. Calculate semi-perimeter ($s$): $s = \frac{120 + 80 + 50}{2} = \frac{250}{2} = 125\text{ m}$
3. Apply Heron's Formula: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ $\text{Area} = \sqrt{125(125-120)(125-80)(125-50)}$ $\text{Area} = \sqrt{125 \times 5 \times 45 \times 75}$ $\text{Area} = \sqrt{(25 \times 5) \times 5 \times (9 \times 5) \times (25 \times 3)}$ $\text{Area} = 25 \times 5 \times 3 \sqrt{15} = 375\sqrt{15}\text{ m}^2$ $\text{Area} \approx 375 \times 3.87 = 1451.25\text{ m}^2$

To find the area for planting grass, we calculate the semi-perimeter first and then substitute the values into Heron's formula. Factoring the numbers inside the square root helps in simplifying the radical.

1. Let the sides be $3x$, $5x$, and $7x$.
2. Given perimeter = $300\text{ m}$: $3x + 5x + 7x = 300$ $15x = 300 \implies x = 20$
3. Find actual side lengths: $a = 3 \times 20 = 60\text{ m}$ $b = 5 \times 20 = 100\text{ m}$ $c = 7 \times 20 = 140\text{ m}$
4. Calculate semi-perimeter ($s$): $s = \frac{300}{2} = 150\text{ m}$
5. Apply Heron's Formula: $\text{Area} = \sqrt{150(150-60)(150-100)(150-140)}$ $\text{Area} = \sqrt{150 \times 90 \times 50 \times 10}$ $\text{Area} = \sqrt{6750000} = 1500\sqrt{3}\text{ m}^2$ $\text{Area} \approx 1500 \times 1.732 = 2598\text{ m}^2$

First, find the individual side lengths using the ratio and the perimeter. Once the side lengths are known, the semi-perimeter and Heron's formula can be used to determine the area of the plot.