Heron’s Formula
Practice Questions – (Level 01)
Q1.
Two sides of a triangle are \(7\text{ cm}\) and \(10\text{ cm}\), and the perimeter is \(26\text{ cm}\). Find the area of the triangle using Heron’s formula.
Q2.
A triangle has sides \(13\), \(14\), and \(15\text{ cm}\). Find its area using Heron’s formula.
Q3.
A triangle has sides \(5\), \(12\), and \(13\text{ cm}\). Find the area using Heron’s formula and verify it with \[ \text{Area} = \frac{1}{2} \times 5 \times 12. \]
Q4.
Two sides of a triangle are \(8\text{ cm}\) and \(9\text{ cm}\). If the semi-perimeter is \(12\text{ cm}\), find the length of the third side and the area of the triangle.
Q5.
The three sides of a triangle are three consecutive integers and the perimeter is \(48\). Find the area using Heron’s formula.
Q6.
A triangle has sides \(11\text{ cm}\), \(13\text{ cm}\), and \(20\text{ cm}\). Check whether a triangle can be formed. If yes, find the area using Heron’s formula.
Practice Questions – Triangular Parks & Fencing (Level -02 A)
Q1.
A triangular park has sides \(90\text{ m}\), \(70\text{ m}\), and \(40\text{ m}\). Find the area of the park using Heron’s formula. The cost of fencing the park is ₹\(18\) per metre. If a gate of width \(2\text{ m}\) is left on one side, find the total fencing cost.
Q2.
A triangular playground has sides \(150\text{ m}\), \(120\text{ m}\), and \(100\text{ m}\). Find the area to be covered with grass. Dhawal wants to fence the entire playground except for a gate of width \(3\text{ m}\). If fencing costs ₹\(25\) per metre, find the total cost.
Q3.
A triangular fishfarm has sides \(60\text{ m}\), \(80\text{ m}\), and \(90\text{ m}\). Calculate the area inside the farm. If fencing wire costs ₹\(12\) per metre and a \(5\text{ m}\) wide gap is left for the gate, find the total fencing cost.
Q4.
A triangular garden has sides \(55\text{ m}\), \(65\text{ m}\), and \(80\text{ m}\). How much area needs to be planted with flowers? Fencing is done with wooden poles at ₹\(30\) per metre. A gate of width \(4\text{ m}\) is kept open. Find the cost of fencing the garden.
Q5.
A triangular farm has sides \(100\text{ m}\), \(120\text{ m}\), and \(160\text{ m}\). Find its area using Heron’s formula. The farmer wants to fence it using barbed wire costing ₹\(22\) per metre. If he leaves a \(6\text{ m}\) opening for a gate, determine the fencing expense.
Q6.
A triangular plot of land has sides \(45\text{ m}\), \(50\text{ m}\), and \(75\text{ m}\). Find the grass-planting area. The cost of fencing is ₹\(14\) per metre. If a \(2.5\text{ m}\) wide gate is left open, calculate the total fencing cost.
Practice Questions — Sides in a Given Ratio (Level 02 B)
Q1.
The sides of a triangular plot are in the ratio \(3:5:7\) and its perimeter is \(360\text{ m}\). Find the area of the triangle.
Q2.
The sides of a triangular field are in the ratio \(4:5:6\) and its perimeter is \(240\text{ m}\). Find the area of the field.
Q3.
The sides of a triangular lawn are in the ratio \(2:3:4\) and its perimeter is \(180\text{ m}\). Find the area to be planted with grass.
Q4.
A triangular plot has its sides in the ratio \(5:7:9\). If the perimeter is \(420\text{ m}\), find the area of the plot.
Q5.
The sides of a triangular park are in the ratio \(1:1:1\) and the perimeter is \(96\text{ m}\). Find the area of the park (an equilateral triangle).
Q6.
The sides of a triangular estate are in the ratio \(6:7:11\) and the perimeter is \(600\text{ m}\). Find the area of the estate.
Higher-Order Hybrid Questions (Level -03)
Hybrid Question 1
A triangular eco-park has its sides in the ratio \(3:4:5\) and its perimeter is \(360\text{ m}\). Inside the park, a triangular flower garden is constructed with sides \(50\text{ m},\ 60\text{ m},\ 70\text{ m}\). Using Heron’s formula, answer the following:
(i) Find the actual lengths of the sides of the eco-park.
(ii) Find the area of the eco-park.
(iii) Find the area of the inner flower garden.
(iv) The remaining area is to be tiled at ₹\(45\) per square metre. Find the total tiling cost.
(v) The boundary of the park is to be fenced at ₹\(28\) per metre, leaving a \(4\text{ m}\) gate. Find the fencing cost.
Hybrid Question 2
A triangular agricultural plot has sides \(120\text{ m},\ 140\text{ m},\ 160\text{ m}\). Inside this plot, a triangular water reservoir is dug with sides in the ratio \(5:6:7\) and perimeter \(180\text{ m}\). Use Heron’s formula to answer the following:
(i) Find the area of the outer agricultural plot.
(ii) Find the actual side lengths of the reservoir and then find its area.
(iii) The reservoir is \(2\text{ m}\) deep. Soil removed is spread on the remaining land at a cost of ₹\(12\) per cubic metre. Find the total cost.
(iv) The plot is fenced at ₹\(30\) per metre, leaving a \(3\text{ m}\) gate. Find the total fencing cost.
