Triangle calculator

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

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 65.73443137182   b = 36   c = 55

Area: T = 990
Perimeter: p = 156.7344313718
Semiperimeter: s = 78.36771568591

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 33.20765703151° = 33°12'24″ = 0.58795639853 rad
Angle ∠ C = γ = 56.79334296849° = 56°47'36″ = 0.99112323415 rad

Height: ha = 30.12112546082
Height: hb = 55
Height: hc = 36

Median: ma = 32.86771568591
Median: mb = 57.87105451849
Median: mc = 45.302176597

Inradius: r = 12.63328431409
Circumradius: R = 32.86771568591

Vertex coordinates: A[55; 0] B[0; 0] C[55; 36]
Centroid: CG[36.66766666667; 12]
Coordinates of the circumscribed circle: U[27.5; 18]
Coordinates of the inscribed circle: I[42.36771568591; 12.63328431409]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 146.7933429685° = 146°47'36″ = 0.58795639853 rad
∠ C' = γ' = 123.2076570315° = 123°12'24″ = 0.99112323415 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 36 ; ; c = 55 ; ; alpha = 90° ; ;

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

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 36**2+55**2 - 2 * 36 * 55 * cos 90° } ; ; a = 65.73 ; ;
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 = 65.73 ; ; b = 36 ; ; c = 55 ; ;

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

p = a+b+c = 65.73+36+55 = 156.73 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 156.73 }{ 2 } = 78.37 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.37 * (78.37-65.73)(78.37-36)(78.37-55) } ; ; T = sqrt{ 980100 } = 990 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 990 }{ 65.73 } = 30.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 990 }{ 36 } = 55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 990 }{ 55 } = 36 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 36**2+55**2-65.73**2 }{ 2 * 36 * 55 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 65.73**2+55**2-36**2 }{ 2 * 65.73 * 55 } ) = 33° 12'24" ; ; gamma = 180° - alpha - beta = 180° - 90° - 33° 12'24" = 56° 47'36" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 990 }{ 78.37 } = 12.63 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 65.73 }{ 2 * sin 90° } = 32.87 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 36**2+2 * 55**2 - 65.73**2 } }{ 2 } = 32.867 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 55**2+2 * 65.73**2 - 36**2 } }{ 2 } = 57.871 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 36**2+2 * 65.73**2 - 55**2 } }{ 2 } = 45.302 ; ;
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