Exam-style CBSE practice questions on Heron's formula (Measuring Space: Perimeter and Area). Try each one first, then reveal the correct answer and a step-by-step explanation. Free, from EduLevel — the AI teacher for CBSE.
The sides of a triangle are 3 cm, 4 cm and 5 cm. Using Heron's formula, its area is:
Explanation: The semi-perimeter is s = (3 + 4 + 5)/2 = 6 cm. By Heron's formula, area = √(6 × (6−3) × (6−4) × (6−5)) = √(6 × 3 × 2 × 1) = √36 = 6 cm².
For a triangle with sides 8 cm, 11 cm and 13 cm, the value of s (semi-perimeter) used in Heron's formula is:
Explanation: The semi-perimeter is half of the full perimeter. Perimeter = 8 + 11 + 13 = 32 cm, so s = 32/2 = 16 cm. Using the full perimeter 32 instead of 16 is a common mistake.
Each side of an equilateral triangle is 4 cm. Using Heron's formula, its area is:
Explanation: Here s = (4 + 4 + 4)/2 = 6 cm, so s minus each side is 6 − 4 = 2 cm. Area = √(6 × 2 × 2 × 2) = √48 = 4√3 cm². This matches the direct formula (√3/4) × 4² = 4√3 cm².
A triangular flower bed in a school garden has sides 6 m, 8 m and 10 m. Its area is:
Explanation: The semi-perimeter is s = (6 + 8 + 10)/2 = 12 m. Area = √(12 × 6 × 4 × 2) = √576 = 24 m². Simply multiplying 6 × 8 = 48 forgets the required factor of 1/2.
A triangular park has sides 5 m, 12 m and 13 m. The cost of planting grass in it at ₹50 per m² is:
Explanation: s = (5 + 12 + 13)/2 = 15, so area = √(15 × 10 × 3 × 2) = √900 = 30 m². Cost = 30 × ₹50 = ₹1500.
The sides of a triangle are in the ratio 3 : 4 : 5 and its perimeter is 36 cm. Its area is:
Explanation: Let the sides be 3x, 4x and 5x, so 12x = 36 and x = 3, giving sides 9 cm, 12 cm and 15 cm. Then s = 36/2 = 18 cm. Area = √(18 × 9 × 6 × 3) = √2916 = 54 cm².
The perimeter of an isosceles triangle is 36 cm and each of its equal sides is 13 cm. Its area is:
Explanation: The base = 36 − 13 − 13 = 10 cm, and s = 36/2 = 18 cm. Area = √(18 × (18−13) × (18−13) × (18−10)) = √(18 × 5 × 5 × 8) = √3600 = 60 cm².
A triangular metal sheet has sides 7 cm, 24 cm and 25 cm. Using Heron's formula, its area is:
Explanation: s = (7 + 24 + 25)/2 = 28 cm. Area = √(28 × 21 × 4 × 3) = √7056 = 84 cm². Multiplying 7 × 24 = 168 without the factor 1/2, or pairing the wrong sides like 1/2 × 24 × 25 = 300, gives the wrong answer.
Each side of an equilateral triangle is 2√3 cm. Its area is:
Explanation: Here s = (3 × 2√3)/2 = 3√3 cm, and s minus each side = 3√3 − 2√3 = √3 cm. Area = √(3√3 × √3 × √3 × √3) = √27 = 3√3 cm². This matches (√3/4) × (2√3)² = (√3/4) × 12 = 3√3 cm².
The sides of a triangle are in the ratio 12 : 17 : 25 and its perimeter is 540 cm. Its area is:
Explanation: Let the sides be 12x, 17x and 25x, so 54x = 540 and x = 10, giving sides 120 cm, 170 cm and 250 cm. Then s = 540/2 = 270 cm. Area = √(270 × 150 × 100 × 20) = √81000000 = 9000 cm². Assuming a right angle and taking 1/2 × 120 × 170 = 10200 is incorrect because this triangle is not right-angled.
The sides of a triangle are 13 cm, 14 cm and 15 cm. The length of the altitude drawn to the side of length 14 cm is:
Explanation: s = (13 + 14 + 15)/2 = 21 cm, so area = √(21 × 8 × 7 × 6) = √7056 = 84 cm². Area is also 1/2 × base × height, so 84 = 1/2 × 14 × h = 7h. Therefore h = 84/7 = 12 cm.
A farmer's field is in the shape of a rhombus with each side 30 m and one diagonal 48 m. The area of the field is:
Explanation: The diagonal divides the rhombus into two equal triangles, each with sides 30 m, 30 m and 48 m. For one triangle, s = (30 + 30 + 48)/2 = 54 m, and area = √(54 × 24 × 24 × 6) = √186624 = 432 m². The rhombus is made of two such triangles, so its area = 2 × 432 = 864 m².
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