Heron's Formula Solved Examples (9 Mathematics)
Heron's formula calculates the area of a triangle using only the lengths of its sides, without needing the height. These examples cover triangles with give
TL;DR: Heron's formula calculates the area of a triangle using only the lengths of its sides, without needing the height. These examples cover triangles with…
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Heron's formula calculates the area of a triangle using only the lengths of its sides, without needing the height. These examples cover triangles with give
Heron's Formula — Solved Numerical Examples (Step by Step)
Example 1: Find the area of a triangle with sides 6 cm, 8 cm, and 10 cm.
Solution: Step 1: Check if this is a valid triangle (sum of any two sides > third side).
6 + 8 = 14 > 10 ✓
8 + 10 = 18 > 6 ✓
6 + 10 = 16 > 8 ✓
Step 2: Calculate semi-perimeter s.
s = (a + b + c) / 2 = (6 + 8 + 10) / 2 = 24 / 2 = 12 cm
Step 3: Apply Heron's formula.
Area = √[s(s-a)(s-b)(s-c)]
Area = √[12(12-6)(12-8)(12-10)]
Area = √[12 × 6 × 4 × 2]
Area = √576 = 24 cm²
Example 2: Find the area of an equilateral triangle with side 4 cm.
Solution: Step 1: Calculate semi-perimeter s.
s = (4 + 4 + 4) / 2 = 12 / 2 = 6 cm
Step 2: Apply Heron's formula.
Area = √[6(6-4)(6-4)(6-4)]
Area = √[6 × 2 × 2 × 2]
Area = √48 = √(16 × 3) = 4√3 cm²
Note: For an equilateral triangle with side a, area = (√3/4)a² = (√3/4)(16) = 4√3
Example 3: Find the area of a triangle with sides 5 cm, 5 cm, and 6 cm.
Solution: Step 1: Calculate semi-perimeter s.
s = (5 + 5 + 6) / 2 = 16 / 2 = 8 cm
Step 2: Apply Heron's formula.
Area = √[8(8-5)(8-5)(8-6)]
Area = √[8 × 3 × 3 × 2]
Area = √144 = 12 cm²
Example 4: The sides of a quadrilateral are 13 cm, 14 cm, 15 cm, and 16 cm. If the diagonal is 15 cm, find its area.
Solution: Step 1: Divide the quadrilateral into two triangles using the diagonal of 15 cm.
Triangle 1: sides 13, 14, 15
s1 = (13 + 14 + 15) / 2 = 21 cm
Area1 = √[21(21-13)(21-14)(21-15)]
Area1 = √[21 × 8 × 7 × 6]
Area1 = √7056 = 84 cm²
Triangle 2: sides 15, 15, 16
s2 = (15 + 15 + 16) / 2 = 23 cm
Area2 = √[23(23-15)(23-15)(23-16)]
Area2 = √[23 × 8 × 8 × 7]
Area2 = √10304 ≈ 101.51 cm²
Total Area = 84 + 101.51 = 185.51 cm² (approximately)
Example 5: A triangle has sides in the ratio 3:4:5. If the perimeter is 24 cm, find its area.
Solution: Step 1: Find the actual sides.
Let sides be 3x, 4x, 5x
Perimeter = 3x + 4x + 5x = 12x = 24
x = 2
Sides are 6 cm, 8 cm, 10 cm
Step 2: Calculate semi-perimeter s.
s = 24 / 2 = 12 cm
Step 3: Apply Heron's formula.
Area = √[12(12-6)(12-8)(12-10)]
Area = √[12 × 6 × 4 × 2]
Area = √576 = 24 cm²
Example 6: Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
Solution: Step 1: Calculate semi-perimeter s.
s = (7 + 8 + 9) / 2 = 24 / 2 = 12 cm
Step 2: Apply Heron's formula.
Area = √[12(12-7)(12-8)(12-9)]
Area = √[12 × 5 × 4 × 3]
Area = √720
Area = √(144 × 5) = 12√5 cm²
Area ≈ 26.83 cm²
Example 7: Find the area of a triangle with sides 13 cm, 14 cm, and 15 cm.
Solution: Step 1: Calculate semi-perimeter s.
s = (13 + 14 + 15) / 2 = 42 / 2 = 21 cm
Step 2: Apply Heron's formula.
Area = √[21(21-13)(21-14)(21-15)]
Area = √[21 × 8 × 7 × 6]
Area = √7056
Area = 84 cm²
Note: 21 × 8 = 168, 7 × 6 = 42, 168 × 42 = 7056 = 84²
Tips
- Heron's formula works for ANY triangle when you know all three sides.
- Always verify the triangle inequality before calculating: sum of any two sides must be greater than the third.
- For quadrilaterals, divide into two triangles using a diagonal and apply Heron's formula to each.
- Semi-perimeter s is half the total perimeter, and (s - a), (s - b), (s - c) represent distances from s to each side.
Frequently Asked Questions
Why is Heron's formula useful?
Heron's formula allows you to find the area of any triangle knowing only the lengths of its three sides. You do not need to know the height or any angles, making it very practical for real-world measurements.
Can Heron's formula be used for all triangles?
Yes, Heron's formula can be used for any valid triangle. You only need the three side lengths. Make sure the sides satisfy the triangle inequality (sum of any two sides > third side).
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