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About Heron's Formula — Class 9 CBSE

Apply Heron's formula to find areas of triangles and quadrilaterals split into triangles. This topic is part of the CBSE Class 9 mathematics syllabus (chapter: Chapter 12). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Heron's Formula — solved examples for Class 9 CBSE

Example 1easy

Heron's Formula is particularly useful for finding the area of a triangle when:
  1. A)The height and base of the triangle are known.
  2. B)The triangle is equilateral and its side length is known.
  3. C)All three side lengths of the triangle are known, but its height is not easily determined.
  4. D)Only two side lengths and the included angle are known.

Step-by-step solution

  1. The standard formula for the area of a triangle is 1/2 × base × height. This requires knowing the height corresponding to a specific base.
  2. Heron's Formula, Area = √[s(s-a)(s-b)(s-c)], allows calculating the area solely from the lengths of the three sides (a, b, c) and the semi-perimeter (s).
  3. Therefore, it is most useful when the height is not readily available or difficult to calculate, but all three side lengths are known.

Answer: All three side lengths of the triangle are known, but its height is not easily determined.

Example 2medium

Find the area of a triangle whose sides are 15 cm, 20 cm, and 25 cm.
  1. A)100 cm²
  2. B)125 cm²
  3. C)150 cm²
  4. D)175 cm²

Step-by-step solution

  1. Let the sides of the triangle be a = 15 cm, b = 20 cm, c = 25 cm.
  2. Calculate the semi-perimeter (s):
    s=(a+b+c)/2=(15+20+25)/2=60/2=30cm.s = (a + b + c) / 2 = (15 + 20 + 25) / 2 = 60 / 2 = 30 cm.
  3. Apply Heron's formula to find the area (A):
    A=√(s(s−a)(s−b)(s−c))A = √(s(s-a)(s-b)(s-c))
  4. A = √(30(30-15)(30-20)(30-25)) = √(30 × 15 × 10 × 5) = √(3 × 10 × 3 × 5 × 10 × 5) = √(9 × 25 × 100) = 3 × 5 × 10 = 150 cm².

Answer: 150 cm²

Example 3hard

A large triangular field has sides measuring 165 meters, 143 meters, and 154 meters. A farmer wishes to calculate its exact area for crop planning. What is the area of this field?
  1. A)9801 m²
  2. B)10164 m²
  3. C)10527 m²
  4. D)11088 m²

Step-by-step solution

  1. Let the sides be a = 165 m, b = 143 m, c = 154 m.
  2. Calculate the semi-perimeter (s): s = (a + b + c) / 2 = (165 + 143 + 154) / 2 = 462 / 2 = 231 m.
  3. Apply Heron's formula: Area = √(s(s-a)(s-b)(s-c))
  4. Area = √(231 × (231-165) × (231-143) × (231-154)) = √(231 × 66 × 88 × 77)
  5. Factorize the terms: 231 = 3 × 7 × 11; 66 = 2 × 3 × 11; 88 = 2³ × 11; 77 = 7 × 11.
  6. Area = √((3×7×11) × (2×3×11) × (2³×11) × (7×11)) = √(2⁴ × 3² × 7² × 11⁴)
  7. Area = 2² × 3 × 7 × 11² = 4 × 3 × 7 × 121 = 12 × 7 × 121 = 84 × 121 = 10164 m².

Answer: 10164 m²

Practice questions on Heron's Formula

  1. Q1.easy

    A triangle has side lengths 7 cm, 8 cm, and 9 cm. What is its semi-perimeter (s)?
    1. A)12 cm
    2. B)24 cm
    3. C)16 cm
    4. D)10 cm
    Show answer

    Answer: 12 cm

    Hint: Remember, the semi-perimeter is half the perimeter of the triangle.

  2. Q2.easy

    Which of the following statements is TRUE about Heron's Formula for the area of a triangle?
    1. A)The term 's' represents the longest side of the triangle.
    2. B)The expression (s-a), (s-b), and (s-c) must all be positive for a valid triangle.
    3. C)Heron's formula can only be used for equilateral triangles.
    4. D)The area calculated using Heron's formula can sometimes be negative.
    Show answer

    Answer: The expression (s-a), (s-b), and (s-c) must all be positive for a valid triangle.

    Hint: Consider the triangle inequality theorem and what it implies for the relationship between the semi-perimeter and each side.

  3. Q3.easy

    Ravi calculated the area of a triangle with sides 5 cm, 12 cm, and 13 cm using Heron's Formula. He found the semi-perimeter 's' as 15 cm. Then, he calculated (s-a) = 10, (s-b) = 3, (s-c) = 2. He incorrectly wrote the area as √(15 × 10 × 3 × 2) = √(900) = 30 cm². Where did Ravi make a mistake?
    1. A)He made a mistake in calculating the semi-perimeter (s).
    2. B)He made a mistake in calculating (s-a), (s-b), or (s-c).
    3. C)He made a mistake in multiplying the terms inside the square root.
    4. D)He made no mistake; the solution is correct.
    Show answer

    Answer: He made no mistake; the solution is correct.

    Hint: Carefully re-calculate each step: semi-perimeter, then each (s-side) term, and finally the product and square root.

  4. Q4.medium

    An isosceles triangle has equal sides of 13 cm each and a base of 24 cm. What is its area?
    1. A)60 cm²
    2. B)72 cm²
    3. C)84 cm²
    4. D)90 cm²
    Show answer

    Answer: 60 cm²

    Hint: For an isosceles triangle, two sides are equal. Calculate the semi-perimeter 's' and then use Heron's formula.

  5. Q5.medium

    The sides of a triangular plot are in the ratio 3:5:7 and its perimeter is 300 m. Find the area of the plot.
    1. A)1000√3 m²
    2. B)1500√3 m²
    3. C)2000√3 m²
    4. D)2500√3 m²
    Show answer

    Answer: 1500√3 m²

    Hint: Use the given ratio and perimeter to find the actual lengths of the sides first, then proceed with Heron's formula.

  6. Q6.medium

    An equilateral triangular field has a perimeter of 180 m. Find the area of the field.
    1. A)600√3 m²
    2. B)750√3 m²
    3. C)800√3 m²
    4. D)900√3 m²
    Show answer

    Answer: 900√3 m²

    Hint: For an equilateral triangle, all sides are equal. Use the perimeter to find the length of one side.

  7. Q7.hard

    The perimeter of a triangular field is 120 meters. If two of its sides are 25 meters and 39 meters, what is the area of the field?
    1. A)360 m²
    2. B)420 m²
    3. C)480 m²
    4. D)540 m²
    Show answer

    Answer: 420 m²

    Hint: First, use the perimeter and the two given sides to find the length of the third side. Then, proceed with Heron's formula.

  8. Q8.hard

    A quadrilateral ABCD has sides AB = 7 cm, BC = 24 cm, CD = 15 cm, and DA = 20 cm. If angle B is 90°, what is the area of the quadrilateral?
    1. A)210 cm²
    2. B)234 cm²
    3. C)252 cm²
    4. D)260 cm²
    Show answer

    Answer: 234 cm²

    Hint: Divide the quadrilateral into two triangles using diagonal AC. One of them is a right-angled triangle, which simplifies finding its area and the diagonal length.

  9. Q9.hard

    An isosceles triangle has a perimeter of 18 cm. The ratio of its equal side to its base is 5:8. What is the area of the triangle?
    1. A)12 cm²
    2. B)15 cm²
    3. C)18 cm²
    4. D)20 cm²
    Show answer

    Answer: 12 cm²

    Hint: Use the given ratio and perimeter to find the actual lengths of the sides first. Remember, an isosceles triangle has two equal sides.

These are 9 of the 60 questions available for Heron's Formula. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.