Triangle calculator ASA

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

Sides: a = 183.062191354   b = 149.9565540708   c = 105

Area: T = 7872.666588717
Perimeter: p = 438.0177454248
Semiperimeter: s = 219.0098727124

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 55° = 0.96599310886 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: ha = 86.01109646503
Height: hb = 105
Height: hc = 149.9565540708

Median: ma = 91.53109567701
Median: mb = 129.0221959554
Median: mc = 158.8880188158

Inradius: r = 35.94768135839
Circumradius: R = 91.53109567701

Vertex coordinates: A[105; 0] B[0; 0] C[105; 149.9565540708]
Centroid: CG[70; 49.9855180236]
Coordinates of the circumscribed circle: U[52.5; 74.9787770354]
Coordinates of the inscribed circle: I[69.05331864161; 35.94768135839]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 125° = 0.96599310886 rad
∠ C' = γ' = 145° = 0.61108652382 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 55° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 55° = 35° ; ;

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

c = 105 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 105 * fraction{ sin 90° }{ sin 35° } = 183.06 ; ;

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 = 105 * fraction{ sin 55° }{ sin 35° } = 149.96 ; ;


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 = 183.06 ; ; b = 149.96 ; ; c = 105 ; ;

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

p = a+b+c = 183.06+149.96+105 = 438.02 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 438.02 }{ 2 } = 219.01 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 219.01 * (219.01-183.06)(219.01-149.96)(219.01-105) } ; ; T = sqrt{ 61978868.17 } = 7872.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7872.67 }{ 183.06 } = 86.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7872.67 }{ 149.96 } = 105 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7872.67 }{ 105 } = 149.96 ; ;

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{ 149.96**2+105**2-183.06**2 }{ 2 * 149.96 * 105 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 183.06**2+105**2-149.96**2 }{ 2 * 183.06 * 105 } ) = 55° ; ; gamma = 180° - alpha - beta = 180° - 90° - 55° = 35° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7872.67 }{ 219.01 } = 35.95 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 183.06 }{ 2 * sin 90° } = 91.53 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 149.96**2+2 * 105**2 - 183.06**2 } }{ 2 } = 91.531 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 105**2+2 * 183.06**2 - 149.96**2 } }{ 2 } = 129.022 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 149.96**2+2 * 183.06**2 - 105**2 } }{ 2 } = 158.88 ; ;
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