Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Right scalene triangle.

Sides: a = 77.42549005891   b = 48.93548059282   c = 60

Area: T = 1468.044417785
Perimeter: p = 186.3659706517
Semiperimeter: s = 93.18798532587

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 39.2° = 39°12' = 0.68441690668 rad
Angle ∠ C = γ = 50.8° = 50°48' = 0.887662726 rad

Height: ha = 37.92217581599
Height: hb = 60
Height: hc = 48.93548059282

Median: ma = 38.71224502946
Median: mb = 64.79770200535
Median: mc = 57.39987389342

Inradius: r = 15.75549526696
Circumradius: R = 38.71224502946

Vertex coordinates: A[60; 0] B[0; 0] C[60; 48.93548059282]
Centroid: CG[40; 16.31216019761]
Coordinates of the circumscribed circle: U[30; 24.46774029641]
Coordinates of the inscribed circle: I[44.24550473304; 15.75549526696]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 140.8° = 140°48' = 0.68441690668 rad
∠ C' = γ' = 129.2° = 129°12' = 0.887662726 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 90° ; ; beta = 39° 12' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 39° 12' = 50° 48' ; ;

2. By using the law of sines, we calculate unknown side a

c = 60 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 60 * fraction{ sin 90° }{ sin 50° 48' } = 77.42 ; ;

3. By using the law of sines, we calculate last unknown side b

fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 60 * fraction{ sin 39° 12' }{ sin 50° 48' } = 48.93 ; ;


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 = 77.42 ; ; b = 48.93 ; ; c = 60 ; ;

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

p = a+b+c = 77.42+48.93+60 = 186.36 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 186.36 }{ 2 } = 93.18 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 93.18 * (93.18-77.42)(93.18-48.93)(93.18-60) } ; ; T = sqrt{ 2155153.71 } = 1468.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1468.04 }{ 77.42 } = 37.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1468.04 }{ 48.93 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1468.04 }{ 60 } = 48.93 ; ;

8. 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{ 48.93**2+60**2-77.42**2 }{ 2 * 48.93 * 60 } ) = 90° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 77.42**2+60**2-48.93**2 }{ 2 * 77.42 * 60 } ) = 39° 12' ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 77.42**2+48.93**2-60**2 }{ 2 * 77.42 * 48.93 } ) = 50° 48' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1468.04 }{ 93.18 } = 15.75 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 77.42 }{ 2 * sin 90° } = 38.71 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.93**2+2 * 60**2 - 77.42**2 } }{ 2 } = 38.712 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 77.42**2 - 48.93**2 } }{ 2 } = 64.797 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.93**2+2 * 77.42**2 - 60**2 } }{ 2 } = 57.399 ; ;
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