Triangle calculator ASA

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

Sides: a = 1849.205457773   b = 2575.443988194   c = 1500

Area: T = 1365833.254379
Perimeter: p = 5924.644445967
Semiperimeter: s = 2962.322222984

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 100° = 1.7455329252 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: ha = 1477.212162952
Height: hb = 1060.666017178
Height: hc = 1821.111100506

Median: ma = 1893.822045088
Median: mb = 1084.692172523
Median: mc = 2112.753272521

Inradius: r = 461.0688428018
Circumradius: R = 1307.585509672

Vertex coordinates: A[1500; 0] B[0; 0] C[-321.1111005057; 1821.111100506]
Centroid: CG[392.9632998314; 607.0377001685]
Coordinates of the circumscribed circle: U[750; 1071.111100506]
Coordinates of the inscribed circle: I[386.8822347897; 461.0688428018]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 80° = 1.7455329252 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 = 45° ; ; beta = 100° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 45° - 100° = 35° ; ;

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

c = 1500 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 1500 * fraction{ sin(45° ) }{ sin (35° ) } = 1849.2 ; ;

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 = 1500 * fraction{ sin(100° ) }{ sin (35° ) } = 2575.44 ; ;


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 = 1849.2 ; ; b = 2575.44 ; ; c = 1500 ; ;

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

p = a+b+c = 1849.2+2575.44+1500 = 5924.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5924.64 }{ 2 } = 2962.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2962.32 * (2962.32-1849.2)(2962.32-2575.44)(2962.32-1500) } ; ; T = sqrt{ 1.866 * 10**{ 12 } } = 1365833.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1365833.25 }{ 1849.2 } = 1477.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1365833.25 }{ 2575.44 } = 1060.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1365833.25 }{ 1500 } = 1821.11 ; ;

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{ 1849.2**2-2575.44**2-1500**2 }{ 2 * 2575.44 * 1500 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2575.44**2-1849.2**2-1500**2 }{ 2 * 1849.2 * 1500 } ) = 100° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1500**2-1849.2**2-2575.44**2 }{ 2 * 2575.44 * 1849.2 } ) = 35° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1365833.25 }{ 2962.32 } = 461.07 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1849.2 }{ 2 * sin 45° } = 1307.59 ; ;




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