Triangle calculator ASA

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

Sides: a = 758.9444408218   b = 1512.062193368   c = 840

Area: T = 196246.1990303
Perimeter: p = 3111.00663419
Semiperimeter: s = 1555.503317095

Angle ∠ A = α = 18° = 0.31441592654 rad
Angle ∠ B = β = 142° = 2.47883675378 rad
Angle ∠ C = γ = 20° = 0.34990658504 rad

Height: ha = 517.1565639274
Height: hb = 259.5744275275
Height: hc = 467.2532834054

Median: ma = 1162.741093931
Median: mb = 263.0888358853
Median: mc = 1120.162246724

Inradius: r = 126.163251382
Circumradius: R = 1227.998784807

Vertex coordinates: A[840; 0] B[0; 0] C[-598.0566355066; 467.2532834054]
Centroid: CG[80.64878816448; 155.7510944685]
Coordinates of the circumscribed circle: U[420; 1153.941051617]
Coordinates of the inscribed circle: I[43.44112372704; 126.163251382]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162° = 0.31441592654 rad
∠ B' = β' = 38° = 2.47883675378 rad
∠ C' = γ' = 160° = 0.34990658504 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 18° ; ; beta = 142° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 18° - 142° = 20° ; ;

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

c = 840 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 840 * fraction{ sin(18° ) }{ sin (20° ) } = 758.94 ; ;

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 = 840 * fraction{ sin(142° ) }{ sin (20° ) } = 1512.06 ; ;


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 = 758.94 ; ; b = 1512.06 ; ; c = 840 ; ;

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

p = a+b+c = 758.94+1512.06+840 = 3111.01 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3111.01 }{ 2 } = 1555.5 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1555.5 * (1555.5-758.94)(1555.5-1512.06)(1555.5-840) } ; ; T = sqrt{ 38512567208.3 } = 196246.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 196246.19 }{ 758.94 } = 517.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 196246.19 }{ 1512.06 } = 259.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 196246.19 }{ 840 } = 467.25 ; ;

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{ 758.94**2-1512.06**2-840**2 }{ 2 * 1512.06 * 840 } ) = 18° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1512.06**2-758.94**2-840**2 }{ 2 * 758.94 * 840 } ) = 142° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 840**2-758.94**2-1512.06**2 }{ 2 * 1512.06 * 758.94 } ) = 20° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 196246.19 }{ 1555.5 } = 126.16 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 758.94 }{ 2 * sin 18° } = 1228 ; ;




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