Heron S Formula — Class 9 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Heron S Formula" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Heron S Formula" — 8 important questions with detailed answers for CBSE board exam…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 9 Mathematics — Heron S Formula. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Heron S Formula — All 8 Questions Solved
Q1. State Heron's formula for the area of a triangle: For a triangle with sides a, b, c, the semi-perimeter s = (a+b+c)/2, and area = √[s(s-a)(s-b)(s-c)]. Find the area of a triangle with sides 5 cm, 6 cm, and 7 cm.
Step 1: Heron's Formula: For a triangle with sides a, b, and c:
Semi-perimeter s = (a + b + c)/2
Area = √[s(s - a)(s - b)(s - c)]
This formula is used when the height is not given but all three sides are known.
Step 2: Given sides:
a = 5 cm
b = 6 cm
c = 7 cm
Step 3: First, verify that these sides can form a triangle using triangle inequality:
5 + 6 = 11 > 7 ✓
6 + 7 = 13 > 5 ✓
5 + 7 = 12 > 6 ✓
Step 4: Calculate semi-perimeter:
s = (a + b + c)/2
s = (5 + 6 + 7)/2
s = 18/2
s = 9 cm
Step 5: Calculate (s - a), (s - b), (s - c):
s - a = 9 - 5 = 4
s - b = 9 - 6 = 3
s - c = 9 - 7 = 2
Step 6: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[9 × 4 × 3 × 2]
Area = √[9 × 24]
Area = √216
Step 7: Simplify √216:
216 = 36 × 6
√216 = √(36 × 6) = 6√6 cm²
Step 8: Numerical value:
√6 ≈ 2.449
Area ≈ 6 × 2.449 ≈ 14.70 cm²
Answer: Area of the triangle = 6√6 cm² ≈ 14.70 cm².
Q2. Using Heron's formula, find the area of a triangle with sides 10 cm, 10 cm, and 12 cm (isosceles triangle).
Step 1: Given sides of isosceles triangle:
a = 10 cm
b = 10 cm
c = 12 cm
Step 2: Verify triangle inequality:
10 + 10 = 20 > 12 ✓
10 + 12 = 22 > 10 ✓
10 + 12 = 22 > 10 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (10 + 10 + 12)/2
s = 32/2
s = 16 cm
Step 4: Calculate the differences:
s - a = 16 - 10 = 6
s - b = 16 - 10 = 6
s - c = 16 - 12 = 4
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[16 × 6 × 6 × 4]
Area = √[16 × 36 × 4]
Area = √[16 × 144]
Area = √2304
Area = 48 cm²
Step 6: Verification using base-height method:
For isosceles triangle with equal sides 10 cm and base 12 cm:
Draw altitude from apex to base, which bisects the base (6 cm each).
In right triangle: 10² = 6² + h²
100 = 36 + h²
h² = 64
h = 8 cm
Area = (1/2) × base × height = (1/2) × 12 × 8 = 48 cm² ✓
Answer: Area of the isosceles triangle = 48 cm².
Q3. Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm using Heron's formula. Express answer in simplest form.
Step 1: Given sides:
a = 13 cm
b = 14 cm
c = 15 cm
Step 2: Verify triangle inequality:
13 + 14 = 27 > 15 ✓
14 + 15 = 29 > 13 ✓
13 + 15 = 28 > 14 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (13 + 14 + 15)/2
s = 42/2
s = 21 cm
Step 4: Calculate the differences:
s - a = 21 - 13 = 8
s - b = 21 - 14 = 7
s - c = 21 - 15 = 6
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[21 × 8 × 7 × 6]
Step 6: Calculate the product:
21 × 8 = 168
7 × 6 = 42
168 × 42 = 7056
Area = √7056
Step 7: Factorize 7056:
7056 = 16 × 441 = 16 × 441
441 = 21²
7056 = 16 × 21²
√7056 = √(16 × 441) = 4 × 21 = 84 cm²
Step 8: Verification by another method:
7056 = 2⁴ × 3² × 7²
√7056 = 2² × 3 × 7 = 4 × 3 × 7 = 84 cm²
Answer: Area of the triangle = 84 cm².
Q4. A triangle has sides in the ratio 3:4:5. If its perimeter is 36 cm, find its area using Heron's formula.
Step 1: Given: Sides in ratio 3:4:5, Perimeter = 36 cm
Step 2: Let the sides be 3x, 4x, and 5x (where x is a constant).
Step 3: Use perimeter to find x:
Perimeter = 3x + 4x + 5x = 36
12x = 36
x = 3
Step 4: Find the actual sides:
Side a = 3x = 3 × 3 = 9 cm
Side b = 4x = 4 × 3 = 12 cm
Side c = 5x = 5 × 3 = 15 cm
Step 5: Verify: 9 + 12 + 15 = 36 ✓
Step 6: Note that 9, 12, 15 are multiples of 3, 4, 5, which is a Pythagorean triplet.
Verification: 9² + 12² = 81 + 144 = 225 = 15²
This is a right triangle with legs 9 cm and 12 cm, and hypotenuse 15 cm.
Step 7: For a right triangle, area can be calculated as:
Area = (1/2) × leg₁ × leg₂ = (1/2) × 9 × 12 = 54 cm²
Step 8: Verify using Heron's formula:
s = (9 + 12 + 15)/2 = 36/2 = 18 cm
Area = √[18 × (18-9) × (18-12) × (18-15)]
Area = √[18 × 9 × 6 × 3]
Area = √[18 × 9 × 18]
Area = √[18² × 9]
Area = 18√9
Area = 18 × 3
Area = 54 cm² ✓
Answer: Area of the triangle = 54 cm².
Q5. Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm. Express the answer both in simplified radical form and decimal form.
Step 1: Given sides:
a = 7 cm
b = 8 cm
c = 9 cm
Step 2: Verify triangle inequality:
7 + 8 = 15 > 9 ✓
8 + 9 = 17 > 7 ✓
7 + 9 = 16 > 8 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (7 + 8 + 9)/2
s = 24/2
s = 12 cm
Step 4: Calculate the differences:
s - a = 12 - 7 = 5
s - b = 12 - 8 = 4
s - c = 12 - 9 = 3
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[12 × 5 × 4 × 3]
Area = √[12 × 60]
Area = √720
Step 6: Simplify √720:
720 = 144 × 5 = 12² × 5
√720 = √(144 × 5) = 12√5 cm²
Step 7: Calculate decimal value:
√5 ≈ 2.236
Area ≈ 12 × 2.236 ≈ 26.83 cm²
Step 8: Verification by factoring:
720 = 16 × 45 = 16 × 9 × 5 = 144 × 5
√720 = √(144 × 5) = 12√5 ✓
Answer: Area = 12√5 cm² ≈ 26.83 cm².
Q6. An equilateral triangle has side length a cm. Derive a formula for its area using Heron's formula and compare with the standard formula (√3/4)a².
Step 1: For an equilateral triangle with side a:
All sides are equal: sides = a, a, a
Step 2: Calculate semi-perimeter:
s = (a + a + a)/2 = 3a/2
Step 3: Calculate the differences:
s - a = 3a/2 - a = a/2
s - a = 3a/2 - a = a/2
s - a = 3a/2 - a = a/2
All three differences equal a/2 (property of equilateral triangles).
Step 4: Apply Heron's formula:
Area = √[s(s - a)(s - a)(s - a)]
Area = √[(3a/2) × (a/2) × (a/2) × (a/2)]
Area = √[(3a/2) × (a/2)³]
Area = √[(3a/2) × (a³/8)]
Area = √[3a⁴/16]
Area = √(3a⁴)/√16
Area = a²√3/4 cm²
Step 5: Compare with standard formula:
Standard formula: Area = (√3/4)a²
Heron's formula result: Area = (√3/4)a²
Both are identical ✓
Step 6: Example with a = 6 cm:
Area = (√3/4) × 6²
Area = (√3/4) × 36
Area = 9√3 cm²
Area ≈ 9 × 1.732 ≈ 15.59 cm²
Answer: Area of equilateral triangle with side a = (√3/4)a² = a²√3/4 cm². Formula derived from Heron's formula matches the standard formula.
Q7. A triangular field has sides 50 m, 60 m, and 70 m. Find its area using Heron's formula. If the cost of fencing is Rs 5 per meter, find the total cost of fencing the field.
Step 1: Given sides of triangular field:
a = 50 m
b = 60 m
c = 70 m
Step 2: Verify triangle inequality:
50 + 60 = 110 > 70 ✓
60 + 70 = 130 > 50 ✓
50 + 70 = 120 > 60 ✓
Step 3: Calculate semi-perimeter:
s = (a + b + c)/2
s = (50 + 60 + 70)/2
s = 180/2
s = 90 m
Step 4: Calculate the differences:
s - a = 90 - 50 = 40
s - b = 90 - 60 = 30
s - c = 90 - 70 = 20
Step 5: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[90 × 40 × 30 × 20]
Area = √[90 × 40 × 30 × 20]
Step 6: Calculate the product:
90 × 40 = 3600
30 × 20 = 600
3600 × 600 = 2,160,000
Area = √2,160,000
Step 7: Simplify:
2,160,000 = 21,600 × 100 = 216 × 10,000
√2,160,000 = √(216 × 10,000) = 100√216 = 100 × 6√6 = 600√6 m²
Step 8: Numerical value:
√6 ≈ 2.449
Area ≈ 600 × 2.449 ≈ 1469.4 m²
Step 9: Calculate perimeter (for fencing):
Perimeter = a + b + c = 50 + 60 + 70 = 180 m
Step 10: Cost of fencing:
Cost per meter = Rs 5
Total cost = Perimeter × Cost per meter
Total cost = 180 × 5 = Rs 900
Answer: Area of the triangular field = 600√6 m² ≈ 1469.4 m²; Total cost of fencing = Rs 900.
Q8. Verify Heron's formula for a right triangle with sides 3 cm, 4 cm, and 5 cm. Compare the area from Heron's formula with the area calculated using the standard right triangle formula (1/2 × base × height).
Step 1: Right triangle with sides:
a = 3 cm (one leg)
b = 4 cm (other leg)
c = 5 cm (hypotenuse)
Step 2: Verify it's a right triangle:
3² + 4² = 9 + 16 = 25 = 5² ✓
This is a Pythagorean triplet.
Step 3: Using standard right triangle formula:
Area = (1/2) × base × height
Area = (1/2) × 3 × 4
Area = (1/2) × 12
Area = 6 cm²
Step 4: Using Heron's formula:
Calculate semi-perimeter:
s = (3 + 4 + 5)/2
s = 12/2
s = 6 cm
Step 5: Calculate the differences:
s - a = 6 - 3 = 3
s - b = 6 - 4 = 2
s - c = 6 - 5 = 1
Step 6: Apply Heron's formula:
Area = √[s(s - a)(s - b)(s - c)]
Area = √[6 × 3 × 2 × 1]
Area = √36
Area = 6 cm²
Step 7: Comparison:
Standard formula result: 6 cm²
Heron's formula result: 6 cm²
Both methods give identical result ✓
Step 8: This demonstrates that Heron's formula is a general formula that works for all triangles, including right triangles, and produces the same result as the specific formula for right triangles.
Answer: Area using Heron's formula = 6 cm²; Area using right triangle formula = 6 cm². Both methods confirm the area is 6 cm².
Heron S Formula — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Find the area of a triangle whose sides are 3 cm, 4 cm, and 5 cm.
- A. 6 cm²
- B. 7.5 cm²
- C. 8 cm²
- D. 12 cm²
Answer: A. 6 cm² — This is a right-angled triangle, and its area can be found directly or by using Heron's formula, both yielding 6 cm².
MCQ 2. What is the semi-perimeter of a triangle with sides a, b, and c?
- A. a+b+c
- B. (a+b+c)/2
- C. √(a+b+c)
- D. a × b × c
Answer: B. (a+b+c)/2 — The semi-perimeter of a triangle is defined as half of its total perimeter (the sum of its three sides).
MCQ 3. Using Heron's formula, find the area of an equilateral triangle with side 6 cm.
- A. 9√3 cm²
- B. 12√3 cm²
- C. 18 cm²
- D. 36 cm²
Answer: A. 9√3 cm² — For an equilateral triangle, all sides are equal. Calculate the semi-perimeter and apply Heron's formula.
MCQ 4. An isosceles triangle has two sides of length 10 cm each and a base of 12 cm. Find its area.
- A. 36 cm²
- B. 48 cm²
- C. 60 cm²
- D. 72 cm²
Answer: B. 48 cm² — For an isosceles triangle with given side lengths, calculate the semi-perimeter and then use Heron's formula to find its area.
MCQ 5. A triangle has a perimeter of 48 m. Two of its sides are 10 m and 17 m. Find its area.
- A. 84 m²
- B. 96 m²
- C. 105 m²
- D. 120 m²
Answer: A. 84 m² — First, use the perimeter to find the length of the third side, then apply Heron's formula to calculate the area.
MCQ 6. The sides of a triangular field are in the ratio 3:5:7 and its perimeter is 300 m. Find the area of the field.
- A. 1500√3 m²
- B. 1000√3 m²
- C. 2000√3 m²
- D. 3000√3 m²
Answer: A. 1500√3 m² — Use the given ratio and perimeter to determine the actual lengths of the sides, then apply Heron's formula to find the area.
MCQ 7. A triangular park has sides 12m, 17m, and 25m. Find the cost of planting grass at Rs 100 per m².
- A. 90 m², Rs 9000
- B. 84 m², Rs 8400
- C. 100 m², Rs 10000
- D. 120 m², Rs 12000
Answer: A. 90 m², Rs 9000 — First, calculate the area of the triangular park using Heron's formula, then multiply the area by the given cost per square meter.
MCQ 8. Find the area of a quadrilateral ABCD where AB = 5 cm, BC = 12 cm, CD = 14 cm, DA = 15 cm and AC = 13 cm.
- A. 104 cm²
- B. 114 cm²
- C. 124 cm²
- D. 134 cm²
Answer: B. 114 cm² — Divide the quadrilateral into two triangles using the diagonal, calculate the area of each triangle using Heron's formula, and then sum the areas.
MCQ 9. Calculate the area of a triangle with sides 8 cm, 15 cm, and 17 cm using Heron's formula.
- A. 60 cm²
- B. 75 cm²
- C. 90 cm²
- D. 100 cm²
Answer: A. 60 cm² — Given the three side lengths, calculate the semi-perimeter and then apply Heron's formula to find the area.
MCQ 10. Heron's formula for the area of a triangle with sides a, b, c and semi-perimeter s is:
- A. √(s(s-a)(s-b)(s-c))
- B. (a+b+c)/2
- C. s × h
- D. (1/2) × base × height
Answer: A. √(s(s-a)(s-b)(s-c)) — Heron's formula is specifically designed to calculate the area of a triangle when only the lengths of its three sides are known.
MCQ 11. An isosceles triangle has a perimeter of 32 cm. The ratio of the equal side to the base is 3:2. Find its area.
- A. 32√2 cm²
- B. 24√2 cm²
- C. 16√3 cm²
- D. 32√3 cm²
Answer: A. 32√2 cm² — Use the given perimeter and the ratio of the side lengths to determine the actual lengths of all three sides, then apply Heron's formula.
MCQ 12. Find the area of an equilateral triangle whose perimeter is 60 cm.
- A. 100√3 cm²
- B. 150√3 cm²
- C. 200√3 cm²
- D. 300√3 cm²
Answer: A. 100√3 cm² — First, determine the length of one side from the perimeter, then use Heron's formula (or the specific formula for equilateral triangles) to find the area.
MCQ 13. An isosceles triangle has a perimeter of 36 cm. Its non-equal side is 6 cm longer than one of the equal sides. Find its area.
- A. 48 cm²
- B. 54 cm²
- C. 60 cm²
- D. 72 cm²
Answer: A. 48 cm² — Set up algebraic equations using the perimeter and the relationship between the side lengths to find the dimensions, then apply Heron's formula.
MCQ 14. If the sides of a triangle are doubled, by what factor does its area increase?
- A. 2
- B. 3
- C. 4
- D. 8
Answer: C. 4 — When the linear dimensions (sides) of a shape are scaled by a factor 'k', its area scales by a factor of k².
MCQ 15. A parallelogram has sides 12 cm and 9 cm and one of its diagonals is 15 cm. Find the area of the parallelogram.
- A. 54 cm²
- B. 72 cm²
- C. 90 cm²
- D. 108 cm²
Answer: D. 108 cm² — A diagonal divides a parallelogram into two congruent triangles. Calculate the area of one triangle using Heron's formula and then double it.
MCQ 16. A rhombus has a perimeter of 52 cm. If one of its diagonals is 10 cm, find its area.
- A. 120 cm²
- B. 144 cm²
- C. 156 cm²
- D. 168 cm²
Answer: A. 120 cm² — A rhombus has four equal sides. Use the perimeter to find the side length. Then, a diagonal divides the rhombus into two congruent triangles, allowing Heron's formula to be applied.
MCQ 17. What is the semi-perimeter of a triangle whose sides are 7 cm, 24 cm, and 25 cm?
- A. 28 cm
- B. 30 cm
- C. 32 cm
- D. 36 cm
Answer: A. 28 cm — The semi-perimeter is found by adding the lengths of all three sides and then dividing the sum by two.
MCQ 18. A triangular board has sides 7 cm, 24 cm, and 25 cm. Find the cost of painting it at a rate of Rs 2 per cm².
- A. Rs 168
- B. Rs 336
- C. Rs 175
- D. Rs 200
Answer: A. Rs 168 — First, calculate the area of the triangular board using Heron's formula, then multiply the area by the given cost per square centimeter.
MCQ 19. A triangle has a perimeter of 24 cm. One side is 10 cm, and the difference between the other two sides is 2 cm. Find its area.
- A. 24 cm²
- B. 30 cm²
- C. 36 cm²
- D. 48 cm²
Answer: A. 24 cm² — Use the given perimeter and the relationship between the two unknown sides to determine all three side lengths, then apply Heron's formula.
MCQ 20. A triangular park has sides 12m, 17m, and 25m. A gardener has to plant grass in it and also fence the boundary. If the cost of planting grass is Rs 100 per m² and the cost of fencing is Rs 20 per meter, what is the total cost?
- A. Rs 10080
- B. Rs 9980
- C. Rs 10180
- D. Rs 9880
Answer: A. Rs 10080 — Calculate the area for the grass planting cost and the perimeter for the fencing cost, then sum these two costs to find the total.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.