MATHEMATICS • QUADRILATERALS
Brahmagupta’s Formula for the Area of a Cyclic 4-gon
Brahmagupta’s formula helps us find the area of a cyclic quadrilateral when the lengths of all four sides are known.
It is similar to Heron’s formula for triangles, but it is used for a special kind of four-sided shape: a cyclic quadrilateral.
What Is a Cyclic Quadrilateral?
A cyclic quadrilateral is a four-sided shape whose four vertices all lie on the same circle. The word “cyclic” refers to the circle passing through all four corners.
An important property is that its opposite angles add up to 180°. Brahmagupta’s formula is specifically for this type of quadrilateral; it does not give the area of every quadrilateral from its side lengths alone.
Brahmagupta’s Formula
Area =
Here:
- a, b, c and d are the four side lengths.
- s is the semiperimeter, which is half the perimeter.
- Area is measured in square units, such as cm² or m².
Remember: the quadrilateral must be cyclic for this formula to give its area from the four side lengths.
Steps to Use the Formula
- Write down the lengths of all four sides.
- Add the sides and divide the total by 2 to find the semiperimeter, s.
- Calculate s − a, s − b, s − c and s − d.
- Multiply these four values.
- Take the square root and write the area in square units.
Solved Example 1: A Cyclic Quadrilateral with Sides 5 cm, 5 cm, 5 cm and 5 cm
Question: Find the area of a cyclic quadrilateral whose four sides are each 5 cm.
Step 1: Find the semiperimeter.
s = (5 + 5 + 5 + 5)/2 = 20/2 = 10 cm
Step 2: Apply Brahmagupta’s formula.
Area = √[(10 − 5)(10 − 5)(10 − 5)(10 − 5)]
Area = √(5 × 5 × 5 × 5) = √625 = 25 cm²
Answer: The area is 25 cm².
In this example, the cyclic quadrilateral is a square, so the result agrees with the usual square-area formula.
Solved Example 2: Sides 4 cm, 5 cm, 7 cm and 8 cm
Question: A cyclic quadrilateral has side lengths 4 cm, 5 cm, 7 cm and 8 cm. Find its area.
Step 1: Find the semiperimeter.
s = (4 + 5 + 7 + 8)/2 = 24/2 = 12 cm
Step 2: Substitute in the formula.
Area = √[(12 − 4)(12 − 5)(12 − 7)(12 − 8)]
Area = √(8 × 7 × 5 × 4)
Area = √1120 = 4√70 cm²
Approximately, Area ≈ 33.47 cm².
Answer: The area is 4√70 cm², approximately 33.47 cm².
This calculation assumes that the given quadrilateral is cyclic.
Solved Example 3: A Cyclic Quadrilateral with Sides 6 m, 8 m, 10 m and 12 m
Question: Find the area of a cyclic quadrilateral whose sides measure 6 m, 8 m, 10 m and 12 m.
Step 1: Calculate the semiperimeter.
s = (6 + 8 + 10 + 12)/2 = 36/2 = 18 m
Step 2: Use Brahmagupta’s formula.
Area = √[(18 − 6)(18 − 8)(18 − 10)(18 − 12)]
Area = √(12 × 10 × 8 × 6)
Area = √5760 = 24√10 m²
Approximately, Area ≈ 75.89 m².
Answer: The area is 24√10 m², approximately 75.89 m².
Real-Life Uses
1. Measuring land and layouts
If a four-sided plot is cyclic and its side lengths are known, the formula can be used to calculate its area. In actual surveying, the shape and measurements must be checked carefully.
2. Design and construction
The formula can help with area calculations for suitable cyclic four-sided panels, layouts and design pieces.
3. Geometry and mathematics problems
It is useful in school geometry and competitive-exam questions where the four side lengths of a cyclic quadrilateral are given.
Brahmagupta’s Formula vs Heron’s Formula
| Feature | Heron’s formula | Brahmagupta’s formula |
|---|---|---|
| Shape | Triangle | Cyclic quadrilateral |
| Number of sides | 3 | 4 |
| Semiperimeter | s = (a + b + c)/2 | s = (a + b + c + d)/2 |
| Area formula | √[s(s − a)(s − b)(s − c)] | √[(s − a)(s − b)(s − c)(s − d)] |
Common Mistakes to Avoid
- Forgetting the cyclic condition: Brahmagupta’s formula is for cyclic quadrilaterals, not every four-sided shape.
- Using the perimeter instead of semiperimeter: Add all four sides, then divide by 2.
- Substituting incorrectly: Calculate each bracket, such as (s − a), carefully.
- Forgetting square units: If the side lengths are in metres, the area is in m².
- Mixing it up with Heron’s formula: Heron’s formula is for triangles; Brahmagupta’s formula is for cyclic quadrilaterals.
Practice Questions
- Find the area of a cyclic quadrilateral with sides 3 cm, 4 cm, 4 cm and 5 cm.
- Find the area of a cyclic quadrilateral with sides 5 cm, 6 cm, 7 cm and 8 cm.
- A square has side 9 cm. Use Brahmagupta’s formula to find its area.
Show answers
1. s = 8 cm; area = √(5 × 4 × 4 × 3) = 4√15 cm² (approximately 15.49 cm²). The given shape must be cyclic.
2. s = 13 cm; area = √(8 × 7 × 6 × 5) = 2√210 cm² (approximately 28.98 cm²). The given shape must be cyclic.
3. s = 18 cm; area = √(9 × 9 × 9 × 9) = 81 cm².
Quick Revision
- Shape: Cyclic quadrilateral, with all four vertices on a circle.
- Semiperimeter: s = (a + b + c + d)/2.
- Formula: Area = √[(s − a)(s − b)(s − c)(s − d)].
- Condition: The quadrilateral must be cyclic.
- Related formula: Heron’s formula is used for triangles.
Frequently Asked Questions
What is Brahmagupta’s formula?
It is a formula for finding the area of a cyclic quadrilateral from its four side lengths.
What is a cyclic 4-gon?
A cyclic 4-gon, also called a cyclic quadrilateral, is a four-sided shape whose four vertices lie on the same circle.
Can I use Brahmagupta’s formula for any quadrilateral?
No. The formula gives the area from the side lengths when the quadrilateral is cyclic. For a general quadrilateral, side lengths alone do not usually determine a unique area.
What is the difference between Brahmagupta’s formula and Heron’s formula?
Heron’s formula finds the area of a triangle from three sides. Brahmagupta’s formula finds the area of a cyclic quadrilateral from four sides.