Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Acute scalene triangle.

Sides: a = 99.42988579437   b = 105.7098868259   c = 50

Area: T = 2467.193325785
Perimeter: p = 255.1387726203
Semiperimeter: s = 127.5698863102

Angle ∠ A = α = 69° = 1.20442771839 rad
Angle ∠ B = β = 83° = 1.44986232792 rad
Angle ∠ C = γ = 28° = 0.48986921906 rad

Height: ha = 49.62773075821
Height: hb = 46.67990213249
Height: hc = 98.68877303141

Median: ma = 66.07331259011
Median: mb = 58.30548684831
Median: mc = 99.52550285624

Inradius: r = 19.34400897199
Circumradius: R = 53.25113617047

Vertex coordinates: A[50; 0] B[0; 0] C[12.11773296331; 98.68877303141]
Centroid: CG[20.70657765444; 32.89659101047]
Coordinates of the circumscribed circle: U[25; 47.01881616337]
Coordinates of the inscribed circle: I[21.86599948422; 19.34400897199]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111° = 1.20442771839 rad
∠ B' = β' = 97° = 1.44986232792 rad
∠ C' = γ' = 152° = 0.48986921906 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 69° ; ; beta = 83° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 69° - 83° = 28° ; ;

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

c = 50 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 50 * fraction{ sin 69° }{ sin 28° } = 99.43 ; ;

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 = 50 * fraction{ sin 83° }{ sin 28° } = 105.71 ; ;
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 = 99.43 ; ; b = 105.71 ; ; c = 50 ; ;

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

p = a+b+c = 99.43+105.71+50 = 255.14 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 255.14 }{ 2 } = 127.57 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 127.57 * (127.57-99.43)(127.57-105.71)(127.57-50) } ; ; T = sqrt{ 6087042.57 } = 2467.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2467.19 }{ 99.43 } = 49.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2467.19 }{ 105.71 } = 46.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2467.19 }{ 50 } = 98.69 ; ;

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{ 105.71**2+50**2-99.43**2 }{ 2 * 105.71 * 50 } ) = 69° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 99.43**2+50**2-105.71**2 }{ 2 * 99.43 * 50 } ) = 83° ; ; gamma = 180° - alpha - beta = 180° - 69° - 83° = 28° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2467.19 }{ 127.57 } = 19.34 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 99.43 }{ 2 * sin 69° } = 53.25 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 105.71**2+2 * 50**2 - 99.43**2 } }{ 2 } = 66.073 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 50**2+2 * 99.43**2 - 105.71**2 } }{ 2 } = 58.305 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 105.71**2+2 * 99.43**2 - 50**2 } }{ 2 } = 99.525 ; ;
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