Area of Quadrilateral Using Heron's Formula
To find the area of any quadrilateral when all four sides and one diagonal are known, divide it into two triangles using the diagonal and apply Heron's formula to each triangle.
This method works for any quadrilateral — regular or irregular — as long as you know the side lengths and at least one diagonal.
What is Area of Quadrilateral Using Heron's Formula?
Method:
Area of Quadrilateral = Area of Triangle 1 + Area of Triangle 2
Where each triangle's area is found using:
Area = √[s(s−a)(s−b)(s−c)]
and s = (a + b + c)/2 is the semi-perimeter of each triangle.
Area of Quadrilateral Using Heron's Formula Formula
Steps:
- Draw one diagonal of the quadrilateral, dividing it into two triangles.
- For Triangle 1: identify three sides (two sides of quadrilateral + diagonal).
- Calculate its area using Heron's formula.
- For Triangle 2: identify three sides (other two sides + same diagonal).
- Calculate its area using Heron's formula.
- Total area = Area of Triangle 1 + Area of Triangle 2.
Solved Examples
Example 1: Example 1: Basic quadrilateral
Problem: Quadrilateral ABCD has AB = 5 cm, BC = 6 cm, CD = 8 cm, DA = 7 cm, and diagonal AC = 9 cm. Find the area.
Solution:
Triangle ABC: sides 5, 6, 9.
- s₁ = (5 + 6 + 9)/2 = 10
- Area₁ = √[10 × 5 × 4 × 1] = √200 = 10√2 ≈ 14.14 cm²
Triangle ACD: sides 9, 8, 7.
- s₂ = (9 + 8 + 7)/2 = 12
- Area₂ = √[12 × 3 × 4 × 5] = √720 = 12√5 ≈ 26.83 cm²
Total area = 14.14 + 26.83 = 40.97 cm²
Answer: Area ≈ 40.97 cm².
Example 2: Example 2: Quadrilateral field
Problem: A field PQRS has PQ = 40 m, QR = 30 m, RS = 50 m, SP = 35 m, and diagonal PR = 45 m. Find the area.
Solution:
Triangle PQR: sides 40, 30, 45.
- s₁ = 115/2 = 57.5
- Area₁ = √[57.5 × 17.5 × 27.5 × 12.5] = √[345703.125] ≈ 587.97 m²
Triangle PRS: sides 45, 50, 35.
- s₂ = 130/2 = 65
- Area₂ = √[65 × 20 × 15 × 30] = √[585000] ≈ 764.85 m²
Total area ≈ 587.97 + 764.85 = 1352.82 m²
Answer: Area ≈ 1352.82 m².
Example 3: Example 3: With right triangle
Problem: ABCD has AB = 3, BC = 4, CD = 12, DA = 13, diagonal BD = 5. Find the area.
Solution:
Triangle ABD: sides 3, 5, 13 → Wait, check: is 3 + 5 > 13? No (8 < 13). These don't form a triangle.
Let's use diagonal AC instead, or recheck. Actually with AB=3, DA=13, diagonal BD=5: triangle ABD has sides 3, 13, 5. 3+5=8 < 13. Invalid.
Use diagonal AC. Triangle ABC: sides 3, 4, AC. Triangle ACD: sides AC, 12, 13.
But AC is not given. Let's reconsider with BD = 5.
Triangle BCD: sides 4, 12, 5. Check: 4+5=9 < 12. Invalid again.
The given dimensions don't form valid triangles with that diagonal. Always verify the triangle inequality before applying Heron's formula.
Example 4: Example 4: Cost of tiling
Problem: A quadrilateral room has sides 8 m, 6 m, 10 m, 9 m and diagonal 12 m. Find cost of tiling at ₹120 per m².
Solution:
Triangle 1: sides 8, 6, 12.
- s₁ = 26/2 = 13
- Area₁ = √[13 × 5 × 7 × 1] = √455 ≈ 21.33 m²
Triangle 2: sides 10, 9, 12.
- s₂ = 31/2 = 15.5
- Area₂ = √[15.5 × 5.5 × 6.5 × 3.5] = √[1939.1875] ≈ 44.04 m²
Total area ≈ 21.33 + 44.04 = 65.37 m²
Cost = 65.37 × 120 ≈ ₹7,844
Answer: Cost ≈ ₹7,844.
Example 5: Example 5: Rhombus as special case
Problem: A rhombus has side 10 cm and one diagonal 16 cm. Find the area using Heron's formula.
Solution:
The diagonal divides the rhombus into two congruent triangles, each with sides 10, 10, 16.
- s = (10 + 10 + 16)/2 = 18
- Area of each triangle = √[18 × 8 × 8 × 2] = √2304 = 48 cm²
- Total area = 2 × 48 = 96 cm²
Answer: Area = 96 cm².
Example 6: Example 6: Trapezium
Problem: Trapezium ABCD: AB = 11, BC = 8, CD = 5, DA = 8, diagonal AC = 10. Find area.
Solution:
Triangle ABC: 11, 8, 10.
- s₁ = 29/2 = 14.5
- Area₁ = √[14.5 × 3.5 × 6.5 × 4.5] = √[1483.6875] ≈ 38.52 cm²
Triangle ACD: 10, 5, 8.
- s₂ = 23/2 = 11.5
- Area₂ = √[11.5 × 1.5 × 6.5 × 3.5] = √[392.4375] ≈ 19.81 cm²
Total ≈ 38.52 + 19.81 = 58.33 cm²
Answer: Area ≈ 58.33 cm².
Real-World Applications
Applications:
- Land surveying: Irregular plots are divided into triangles for area calculation.
- Construction: Calculating floor areas of non-rectangular rooms.
- Agriculture: Measuring irregular field areas for crop planning.
Key Points to Remember
- To find area of any quadrilateral: divide into two triangles using a diagonal.
- Apply Heron's formula to each triangle separately.
- Total area = sum of both triangle areas.
- You need to know all four sides and one diagonal.
- Always verify the triangle inequality before applying the formula.
- For special quadrilaterals (rhombus, rectangle), simpler formulas exist but Heron's method always works.
- The same quadrilateral can be split using either diagonal — the total area will be the same.
Practice Problems
- Quadrilateral ABCD: AB = 6, BC = 8, CD = 10, DA = 12, diagonal AC = 14. Find area.
- A plot has sides 40, 30, 50, 35 m and diagonal 45 m. Find area and cost of levelling at ₹15/m².
- A rhombus has side 13 cm and diagonal 24 cm. Find area using Heron's formula.
- Verify: for a square with side 10, Heron's method (splitting by diagonal) gives the same area as side².
- A quadrilateral has sides 5, 7, 9, 11 and diagonal 10. Find its area.
Frequently Asked Questions
Q1. Can you find the area of any quadrilateral using Heron's formula?
Yes, if you know all four sides and at least one diagonal. Divide into two triangles and apply Heron's formula to each.
Q2. What if the diagonal length is not given?
You cannot directly apply this method. You would need additional information (angles, perpendicular heights, or coordinates) to find the area.
Q3. Does it matter which diagonal you use?
No. The total area is the same regardless of which diagonal you choose, as long as it lies inside the quadrilateral.
Q4. What is the triangle inequality?
For three sides to form a triangle, the sum of any two sides must be greater than the third. If this fails, Heron's formula cannot be applied.
Q5. Is this method used in real land surveys?
Yes. Surveyors routinely divide irregular plots into triangles and use this method (or coordinate geometry) to calculate areas.
