Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle γ.

Right isosceles triangle.

Sides: a = 16.5   b = 16.5   c = 23.33545237792

Area: T = 136.125
Perimeter: p = 56.33545237792
Semiperimeter: s = 28.16772618896

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 16.5
Height: hb = 16.5
Height: hc = 11.66772618896

Median: ma = 18.44875608144
Median: mb = 18.44875608144
Median: mc = 11.66772618896

Inradius: r = 4.83327381104
Circumradius: R = 11.66772618896

Vertex coordinates: A[23.33545237792; 0] B[0; 0] C[11.66772618896; 11.66772618896]
Centroid: CG[11.66772618896; 3.88990872965]
Coordinates of the circumscribed circle: U[11.66772618896; -0]
Coordinates of the inscribed circle: I[11.66772618896; 4.83327381104]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 16.5 ; ; b = 16.5 ; ; gamma = 90° ; ;

2. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 16.5**2+16.5**2 - 2 * 16.5 * 16.5 * cos(90° ) } ; ; c = 23.33 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 16.5 ; ; b = 16.5 ; ; c = 23.33 ; ;

3. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 16.5+16.5+23.33 = 56.33 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56.33 }{ 2 } = 28.17 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.17 * (28.17-16.5)(28.17-16.5)(28.17-23.33) } ; ; T = sqrt{ 18530.02 } = 136.13 ; ;

6. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 136.13 }{ 16.5 } = 16.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 136.13 }{ 16.5 } = 16.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 136.13 }{ 23.33 } = 11.67 ; ;

7. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 16.5**2-16.5**2-23.33**2 }{ 2 * 16.5 * 23.33 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.5**2-16.5**2-23.33**2 }{ 2 * 16.5 * 23.33 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23.33**2-16.5**2-16.5**2 }{ 2 * 16.5 * 16.5 } ) = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 136.13 }{ 28.17 } = 4.83 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.5 }{ 2 * sin 45° } = 11.67 ; ;




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