Heron's Formula Word Problems
Heron's Formula calculates the area of a triangle when all three sides are known, without needing the height. This makes it ideal for real-world problems where measuring height is difficult — such as land plots, gardens, and irregular surfaces.
Word problems on Heron's formula typically involve finding areas of triangular regions, calculating costs, or finding areas of quadrilaterals by splitting into triangles.
What is Heron's Formula Word Problems?
Area = √[s(s−a)(s−b)(s−c)]
Where:
- a, b, c are the three sides of the triangle
- s = (a + b + c) / 2 is the semi-perimeter
Heron's Formula Word Problems Formula
Steps to apply Heron's Formula:
- Identify the three sides: a, b, c.
- Calculate semi-perimeter: s = (a + b + c) / 2.
- Calculate s − a, s − b, s − c.
- Compute Area = √[s(s−a)(s−b)(s−c)].
Solved Examples
Example 1: Example 1: Triangular park
Problem: A triangular park has sides 40 m, 32 m, and 24 m. Find the cost of planting grass at ₹10 per m².
Solution:
- s = (40 + 32 + 24)/2 = 96/2 = 48
- Area = √[48 × 8 × 16 × 24] = √[48 × 8 × 16 × 24]
- = √[147456] = 384 m²
- Cost = 384 × 10 = ₹3,840
Answer: Area = 384 m². Cost = ₹3,840.
Example 2: Example 2: Triangular land plot
Problem: A farmer's field is shaped as a triangle with sides 50 m, 60 m, and 70 m. Find its area.
Solution:
- s = (50 + 60 + 70)/2 = 90
- Area = √[90 × 40 × 30 × 20] = √[2160000] = 1469.69 m²
Answer: Area ≈ 1469.69 m².
Example 3: Example 3: Equilateral triangle
Problem: Find the area of an equilateral triangle with side 12 cm using Heron's formula.
Solution:
- s = (12 + 12 + 12)/2 = 18
- Area = √[18 × 6 × 6 × 6] = √[3888] = 36√3 ≈ 62.35 cm²
Answer: Area = 36√3 ≈ 62.35 cm².
Example 4: Example 4: Isosceles triangle
Problem: An isosceles triangle has equal sides of 10 cm and base 12 cm. Find its area.
Solution:
- s = (10 + 10 + 12)/2 = 16
- Area = √[16 × 6 × 6 × 4] = √[2304] = 48 cm²
Answer: Area = 48 cm².
Example 5: Example 5: Cost of fencing and planting
Problem: A triangular garden has sides 26 m, 28 m, and 30 m. Find (a) the cost of fencing at ₹15 per m, and (b) the cost of planting at ₹8 per m².
Solution:
(a) Perimeter = 26 + 28 + 30 = 84 m. Fencing cost = 84 × 15 = ₹1,260.
(b) s = 84/2 = 42. Area = √[42 × 16 × 14 × 12] = √[112896] = 336 m².
Planting cost = 336 × 8 = ₹2,688.
Answer: (a) ₹1,260. (b) ₹2,688.
Example 6: Example 6: Right triangle verification
Problem: Verify using Heron's formula that the area of a right triangle with sides 3, 4, 5 equals ½ × 3 × 4 = 6 cm².
Solution:
- s = (3 + 4 + 5)/2 = 6
- Area = √[6 × 3 × 2 × 1] = √36 = 6 cm²
This matches ½ × 3 × 4 = 6. Verified ✓
Example 7: Example 7: Finding height from area
Problem: A triangle has sides 13, 14, 15. Using Heron's formula, find the area and then the height corresponding to side 14.
Solution:
- s = (13+14+15)/2 = 21
- Area = √[21 × 8 × 7 × 6] = √[7056] = 84 cm²
- Area = ½ × base × height → 84 = ½ × 14 × h → h = 12 cm
Answer: Area = 84 cm². Height = 12 cm.
Example 8: Example 8: Triangular signboard
Problem: A triangular signboard has sides 35 cm, 54 cm, and 61 cm. Find the cost of painting it on both sides at ₹2 per cm².
Solution:
- s = (35+54+61)/2 = 75
- Area = √[75 × 40 × 21 × 14] = √[882000] ≈ 939.15 cm²
- Both sides area = 2 × 939.15 = 1878.3 cm²
- Cost = 1878.3 × 2 = ₹3,756.60
Answer: Cost ≈ ₹3,757.
Example 9: Example 9: Triangular flower bed
Problem: A triangular flower bed has sides 6 m, 8 m, and 10 m. How many bags of fertiliser are needed if one bag covers 12 m²?
Solution:
- Note: 6² + 8² = 36 + 64 = 100 = 10². Right triangle!
- Area = ½ × 6 × 8 = 24 m²
- Bags needed = 24/12 = 2
Answer: 2 bags of fertiliser.
Example 10: Example 10: Perimeter given, side relationship
Problem: The perimeter of a triangle is 60 cm. The sides are in the ratio 3:4:5. Find the area.
Solution:
- Sides: 3x + 4x + 5x = 60 → 12x = 60 → x = 5
- Sides: 15, 20, 25 cm
- s = 30
- Area = √[30 × 15 × 10 × 5] = √[22500] = 150 cm²
Answer: Area = 150 cm².
Real-World Applications
Applications:
- Land surveying: Measuring areas of triangular land plots.
- Construction: Calculating material for triangular surfaces.
- Navigation: Triangulation methods use areas of triangles.
- Art and design: Computing areas of triangular patterns.
Key Points to Remember
- Heron's formula: Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2.
- No height is needed — only the three sides.
- Always compute s first, then s−a, s−b, s−c, and check they are all positive.
- If any of s−a, s−b, s−c is zero or negative, the sides do not form a valid triangle.
- For right triangles, Heron's formula gives the same result as ½ × base × height.
- To find height: use Area = ½ × base × height after finding area by Heron's formula.
- For cost problems: Cost = Area × Rate per unit area.
- The formula works for ALL triangles — scalene, isosceles, equilateral.
Practice Problems
- A triangular plot has sides 120 m, 150 m, and 200 m. Find its area.
- Find the area of a triangle with sides 17, 25, and 26 cm.
- A triangular garden with sides 10, 10, and 12 m needs fencing at ₹20/m and grass at ₹5/m². Find total cost.
- The sides of a triangle are in ratio 5:12:13 and perimeter is 90 cm. Find the area.
- A triangular signboard has sides 20, 21, and 29 cm. Find area and the height to the longest side.
- Check whether sides 3, 4, 8 form a triangle. If yes, find area by Heron's formula.
Frequently Asked Questions
Q1. When should you use Heron's formula?
When all three sides are known but the height is not given. If the height is known, ½ × base × height is simpler.
Q2. What is the semi-perimeter?
Half of the perimeter: s = (a + b + c)/2. It simplifies the formula.
Q3. Can Heron's formula give a negative result?
No. If the three sides form a valid triangle, s(s−a)(s−b)(s−c) is always positive. If negative, the sides do not form a triangle.
Q4. Does Heron's formula work for right triangles?
Yes. It works for all triangles. For a right triangle with legs a, b and hypotenuse c, it gives the same result as ½ab.
Q5. How do you check if three sides form a triangle?
Triangle inequality: sum of any two sides must be greater than the third. Also, in Heron's formula, s−a, s−b, s−c must all be positive.
Q6. Can Heron's formula find the height?
Indirectly. Find area using Heron's formula, then use Area = ½ × base × height to find the height corresponding to any side.
