Is it cyclic?
Rectangles and isosceles trapezia are cyclic
Which shapes are cyclic?
Brahmagupta's formula needs a cyclic 4-gon, one whose four corners lie on a circle. Do shapes we already know have this property? Think of a rectangle, and of an isosceles trapezium, a trapezium whose two slanted sides are equal.
Can one circle pass through all four corners of a rectangle?
What this lesson covers
The idea
All rectangles and all isosceles trapezia are cyclic quadrilaterals, so Brahmagupta's formula applies to them.
Which shapes are cyclic?
Brahmagupta's formula needs a cyclic 4-gon, one whose four corners lie on a circle. Do shapes we already know have this property? Think of a rectangle, and of an isosceles trapezium, a trapezium whose two slanted sides are equal.
Can one circle pass through all four corners of a rectangle?
- Yes, for every rectangle
- Only for a square
- No, never
Test the shapes
Pick a shape and tap Test it. A circle is drawn through A, B and C. Does it pass through D too? Then Brahmagupta's formula is tried. Test all three shapes.
The formula still works
All rectangles and all isosceles trapezia are cyclic quadrilaterals, so Brahmagupta's formula applies to them.
For a rectangle with sides a and b, s = a + b. The formula gives √((a + b − a)(a + b − b)(a + b − a)(a + b − b)) = √(b × a × b × a) = ab, which is correct. For an isosceles trapezium the formula gives the same area as the usual one: half the sum of the parallel sides times the distance between them.
Notes
All rectangles and all isosceles trapezia are cyclic quadrilaterals, so Brahmagupta's formula applies to them.
Check yourself
A rectangle has sides 6 and 4, so s = 10. Use Brahmagupta's formula. What is the area?
Answer: 24
Area = √(4 × 6 × 4 × 6) = √576 = 24, which is 6 × 4.
Can Brahmagupta's formula be used for an isosceles trapezium?
An isosceles trapezium has parallel sides 10 and 4 and two slanted sides of 5 each, so s = 12. What is its area?
Answer: 28
Area = √(2 × 7 × 8 × 7) = √784 = 28.
An isosceles trapezium has parallel sides 14 and 6 and two slanted sides of 5 each, so s = 15. What is its area?
Answer: 30
Area = √(1 × 10 × 9 × 10) = √900 = 30.
- Yes, every isosceles trapezium is cyclic — correct. Yes. Its four corners always lie on one circle.
- No, only rectangles are cyclic. Isosceles trapezia are cyclic too.
- No, it first needs a diagonal. Cyclic 4-gons need only their four sides for the formula.