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Area of a triangle by Heron's formula: Class 9 practice

When all three sides are known but not the height, area = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.. Read the method, follow the three worked examples, then try the six questions.

What you should know first

  • Square roots
  • Perimeter

The idea behind area of a triangle by heron's formula

When all three sides are known but not the height, area = √[s(s − a)(s − b)(s − c)], where s is half the perimeter.

Step-by-step method

  1. s = (a + b + c) ÷ 2.
  2. Find s − a, s − b, s − c.
  3. Multiply s(s − a)(s − b)(s − c).
  4. Take the square root (units²).

Worked example 1 (Basic)

Find the area of a triangle with sides 6 cm, 8 cm and 10 cm.

  1. s = (6 + 8 + 10) ÷ 2 = 12
  2. s − a = 6, s − b = 4, s − c = 2
  3. Area = √(12 × 6 × 4 × 2) = √576
  4. Area = 24 cm²
  5. Answer: 24 cm²

Worked example 2 (Standard)

Find the area of a triangle with sides 20 cm, 26 cm and 26 cm.

  1. s = (20 + 26 + 26) ÷ 2 = 36
  2. s − a = 16, s − b = 10, s − c = 10
  3. Area = √(36 × 16 × 10 × 10) = √57600
  4. Area = 240 cm²
  5. Answer: 240 cm²

Worked example 3 (Challenge)

Find the area of a triangle with sides 14 cm, 30 cm and 40 cm.

  1. s = (14 + 30 + 40) ÷ 2 = 42
  2. s − a = 28, s − b = 12, s − c = 2
  3. Area = √(42 × 28 × 12 × 2) = √28224
  4. Area = 168 cm²
  5. Answer: 168 cm²

Common mistakes and misconceptions

  • Using the full perimeter instead of s
  • Forgetting square units

Practice questions

  1. (Basic) Find the area of a triangle with sides 18 cm, 24 cm and 30 cm.
  2. (Basic) Find the area of a triangle with sides 9 cm, 12 cm and 15 cm.
  3. (Standard) Find the area of a triangle with sides 10 cm, 24 cm and 26 cm.
  4. (Standard) Find the area of a triangle with sides 26 cm, 28 cm and 30 cm.
  5. (Challenge) Find the area of a triangle with sides 18 cm, 20 cm and 34 cm.
  6. (Challenge) Find the area of a triangle with sides 22 cm, 26 cm and 40 cm.

Answer key with full working

  • Q1: s = (18 + 24 + 30) ÷ 2 = 36 → s − a = 18, s − b = 12, s − c = 6 → Area = √(36 × 18 × 12 × 6) = √46656 → Area = 216 cm² → Answer: 216 cm²
  • Q2: s = (9 + 12 + 15) ÷ 2 = 18 → s − a = 9, s − b = 6, s − c = 3 → Area = √(18 × 9 × 6 × 3) = √2916 → Area = 54 cm² → Answer: 54 cm²
  • Q3: s = (10 + 24 + 26) ÷ 2 = 30 → s − a = 20, s − b = 6, s − c = 4 → Area = √(30 × 20 × 6 × 4) = √14400 → Area = 120 cm² → Answer: 120 cm²
  • Q4: s = (26 + 28 + 30) ÷ 2 = 42 → s − a = 16, s − b = 14, s − c = 12 → Area = √(42 × 16 × 14 × 12) = √112896 → Area = 336 cm² → Answer: 336 cm²
  • Q5: s = (18 + 20 + 34) ÷ 2 = 36 → s − a = 18, s − b = 16, s − c = 2 → Area = √(36 × 18 × 16 × 2) = √20736 → Area = 144 cm² → Answer: 144 cm²
  • Q6: s = (22 + 26 + 40) ÷ 2 = 44 → s − a = 22, s − b = 18, s − c = 4 → Area = √(44 × 22 × 18 × 4) = √69696 → Area = 264 cm² → Answer: 264 cm²

Syllabus note

This skill is listed under "Heron's Formula" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

Frequently asked questions

Which chapter is area of a triangle by heron's formula in?

"Heron's Formula", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

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