Triangle calculator ASA

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

Sides: a = 87.47552959688   b = 48.4888348334   c = 53

Area: T = 1123.835492302
Perimeter: p = 188.9643644303
Semiperimeter: s = 94.48218221514

Angle ∠ A = α = 119° = 2.07769418099 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 32° = 0.55985053606 rad

Height: ha = 25.69549098731
Height: hb = 46.35548444784
Height: hc = 42.40988650196

Median: ma = 25.82878553286
Median: mb = 68.13772418093
Median: mc = 65.56988467529

Inradius: r = 11.89547210948
Circumradius: R = 50.00876177422

Vertex coordinates: A[53; 0] B[0; 0] C[76.50876177422; 42.40988650196]
Centroid: CG[43.16992059141; 14.13662883399]
Coordinates of the circumscribed circle: U[26.5; 42.40988650196]
Coordinates of the inscribed circle: I[45.99334738174; 11.89547210948]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 61° = 2.07769418099 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 148° = 0.55985053606 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 119° ; ; beta = 29° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 119° - 29° = 32° ; ;

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

c = 53 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 53 * fraction{ sin 119° }{ sin 32° } = 87.48 ; ;

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 = 53 * fraction{ sin 29° }{ sin 32° } = 48.49 ; ;
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 = 87.48 ; ; b = 48.49 ; ; c = 53 ; ;

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

p = a+b+c = 87.48+48.49+53 = 188.96 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 188.96 }{ 2 } = 94.48 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.48 * (94.48-87.48)(94.48-48.49)(94.48-53) } ; ; T = sqrt{ 1263004.93 } = 1123.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1123.83 }{ 87.48 } = 25.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1123.83 }{ 48.49 } = 46.35 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1123.83 }{ 53 } = 42.41 ; ;

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{ 48.49**2+53**2-87.48**2 }{ 2 * 48.49 * 53 } ) = 119° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 87.48**2+53**2-48.49**2 }{ 2 * 87.48 * 53 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 119° - 29° = 32° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1123.83 }{ 94.48 } = 11.89 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 87.48 }{ 2 * sin 119° } = 50.01 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.49**2+2 * 53**2 - 87.48**2 } }{ 2 } = 25.828 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 53**2+2 * 87.48**2 - 48.49**2 } }{ 2 } = 68.137 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.49**2+2 * 87.48**2 - 53**2 } }{ 2 } = 65.569 ; ;
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