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.

✓ 100% Free ✓ No Login Needed ✓ NCERT / CBSE Aligned ✓ Download as PDF

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 Jul 23, 2026

🤖 Stuck on any question? Ask Syllab's free AI Tutor for a step-by-step explanation — instant, unlimited, no login.

Key Questions Covered:

State Heron's formula for the area of a triangle: For a triangle with sides a…
Using Heron's formula, find the area of a triangle with sides 10 cm, 10 cm, a…
Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm using Heron's …
A triangle has sides in the ratio 3:4:5. If its perimeter is 36 cm, find its …
Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm. Express the answ…
An equilateral triangle has side length a cm. Derive a formula for its area u…
+ 2 more questions in the full chapter

Solutions Summary:

Question	Status
State Heron's formula for the area of a triangle: For a t…	✓ Solved
Using Heron's formula, find the area of a triangle with s…	✓ Solved
Find the area of a triangle with sides 13 cm, 14 cm, and …	✓ Solved
A triangle has sides in the ratio 3:4:5. If its perimeter…	✓ Solved
Find the area of a triangle with sides 7 cm, 8 cm, and 9 …	✓ Solved
An equilateral triangle has side length a cm. Derive a fo…	✓ Solved

Showing 6 of 8 questions

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 St...

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] A...

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...

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 ...

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...

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...

Showing 6 of 8 questions. Visit the full page for complete solutions.