Triangle calculator ASA

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

Sides: a = 328.9587952376   b = 280.5098821904   c = 145

Area: T = 20225.4821979
Perimeter: p = 754.467677428
Semiperimeter: s = 377.233338714

Angle ∠ A = α = 96° = 1.67655160819 rad
Angle ∠ B = β = 58° = 1.01222909662 rad
Angle ∠ C = γ = 26° = 0.45437856055 rad

Height: ha = 122.9676973943
Height: hb = 144.2065674828
Height: hc = 278.9722165228

Median: ma = 151.0032536321
Median: mb = 212.0099121087
Median: mc = 296.9733091035

Inradius: r = 53.6155302008
Circumradius: R = 165.3854972371

Vertex coordinates: A[145; 0] B[0; 0] C[174.3211156087; 278.9722165228]
Centroid: CG[106.4440385362; 92.99107217426]
Coordinates of the circumscribed circle: U[72.5; 148.6477028514]
Coordinates of the inscribed circle: I[96.72545652357; 53.6155302008]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 84° = 1.67655160819 rad
∠ B' = β' = 122° = 1.01222909662 rad
∠ C' = γ' = 154° = 0.45437856055 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 96° ; ; beta = 58° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 96° - 58° = 26° ; ;

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

c = 145 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 145 * fraction{ sin(96° ) }{ sin (26° ) } = 328.96 ; ;

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 = 145 * fraction{ sin(58° ) }{ sin (26° ) } = 280.51 ; ;


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 = 328.96 ; ; b = 280.51 ; ; c = 145 ; ;

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

p = a+b+c = 328.96+280.51+145 = 754.47 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 754.47 }{ 2 } = 377.23 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 377.23 * (377.23-328.96)(377.23-280.51)(377.23-145) } ; ; T = sqrt{ 409070121.28 } = 20225.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20225.48 }{ 328.96 } = 122.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20225.48 }{ 280.51 } = 144.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20225.48 }{ 145 } = 278.97 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 328.96**2-280.51**2-145**2 }{ 2 * 280.51 * 145 } ) = 96° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 280.51**2-328.96**2-145**2 }{ 2 * 328.96 * 145 } ) = 58° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 145**2-328.96**2-280.51**2 }{ 2 * 280.51 * 328.96 } ) = 26° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20225.48 }{ 377.23 } = 53.62 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 328.96 }{ 2 * sin 96° } = 165.38 ; ;




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