Brahmagupta's Formula

A formula for calculating the area of an inscribed, or cyclic quadrilateral

when you know the lengths (a,b,c,d) of the sides.

when you know the lengths (a,b,c,d) of the sides.

Try this Drag any orange dot. Note the formula changes to calculate the area.

Recall that an inscribed (or 'cyclic') quadrilateral is one where the four vertices all lie on a circle. Using the formula below, you can calculate the area of the quadrilateral.

where a,b,c,d are the side lengths, and p is half the perimeter:In the figure above, drag any vertex around the circle. Note how the semi-perimeter (p) and the area are calculated.

In the figure above, if you drag a point past its neighbor the quadrilateral will become 'crossed' where one side crossed over another. In such 'crossed' quadrilaterals the area formula no longer holds. (Most properties of polygons are invalid when the polygon is crossed).

Recall that Heron's formula for the area of a triangle is where p is half the perimeter, as here.

The two formulae are very similar. If you take the Brahmagupta's formula and set d (the length of the fourth side) to zero, the quadrilateral becomes a triangle. In Brahmagupta's formula, the term (p-d) becomes just p and the formulae are then same also.

In the figure above, if you are careful, you can drag the point D around to be on top of A making d zero, illustrating this similarity.

From this you can see that Heron's formula is just a special case of Brahmagupta's. Recall too that all triangles are cyclic. That is, you can always draw a circle through the three vertices. See Circumcircle of a triangle.

In the figure above

- Click on 'Hide details'.
- Drag the vertices around to create a new (uncrossed) quadrilateral.
- Calculate the area of the quadrilateral using Brahmagupta's Formula.
- Estimate the area by counting the squares. Each is one square unit.
- Click "show details" to check your answer.

- Polygon general definition
- Quadrilateral
- Regular polygon
- Irregular polygon
- Convex polygons
- Concave polygons
- Polygon diagonals
- Polygon triangles
- Apothem of a regular polygon
- Polygon center
- Radius of a regular polygon
- Incircle of a regular polygon
- Incenter of a regular polygon
- Circumcircle of a polygon
- Parallelogram inscribed in a quadrilateral

- Square
- Diagonals of a square
- Rectangle
- Diagonals of a rectangle
- Golden rectangle
- Parallelogram
- Rhombus
- Trapezoid
- Trapezoid median
- Kite
- Inscribed (cyclic) quadrilateral

- Regular polygon area
- Irregular polygon area
- Rhombus area
- Kite area
- Rectangle area
- Area of a square
- Trapezoid area
- Parallelogram area

- Perimeter of a polygon (regular and irregular)
- Perimeter of a triangle
- Perimeter of a rectangle
- Perimeter of a square
- Perimeter of a parallelogram
- Perimeter of a rhombus
- Perimeter of a trapezoid
- Perimeter of a kite

- Exterior angles of a polygon
- Interior angles of a polygon
- Relationship of interior/exterior angles
- Polygon central angle

- Tetragon, 4 sides
- Pentagon, 5 sides
- Hexagon, 6 sides
- Heptagon, 7 sides
- Octagon, 8 sides
- Nonagon Enneagon, 9 sides
- Decagon, 10 sides
- Undecagon, 11 sides
- Dodecagon, 12 sides

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