Area of a Triangle
The symbol Δ is also used to represent the area of a triangle.
The basic formula for the area of a triangle is: Δ = (1/2) × base × altitude
For a right-angled triangle, the right angle (90°) is contained by the base and altitude. So if a, b, c are its sides and 'c' is the hypotenuse, then:
Area of a right-angled triangle,
(i) Δ = (1/2) ab
Also (ii) Δ = (1/2) bc sin A = (1/2) ca sin B = (1/2) ab sin C
(iii) Δ =
Note that formula (iii) for the area of a triangle is valid for any triangle.
(iv) Δ = 2R2 sin A sin B sin C
Circumcenter and circumradius:
In a triangle, the point of concurrence of the perpendicular bisectors of the sides is called as circumcenter (represented by 'O'). The circumcenter is equidistant from the three vertices.
i.e., OA = OB = OC = R (say)
So if we draw a circle with 'O' as center and R as radius, it passes through the three vertices. The circle is called circumcircle and 'R' is called as circumradius.
The area of a triangle in terms of 'a', 'b', 'c and 'R' is: Δ = .