Triangle calculator ASA

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

Sides: a = 2597.375515938   b = 2507.07112321   c = 2994

Area: T = 3093016.15995
Perimeter: p = 8098.446639148
Semiperimeter: s = 4049.223319574

Angle ∠ A = α = 55.5° = 55°30' = 0.96986577349 rad
Angle ∠ B = β = 52.7° = 52°42' = 0.92197885158 rad
Angle ∠ C = γ = 71.8° = 71°48' = 1.25331464029 rad

Height: ha = 2381.648760168
Height: hb = 2467.434380873
Height: hc = 2066.143305912

Median: ma = 2436.828819497
Median: mb = 2506.76599244
Median: mc = 2067.577658642

Inradius: r = 763.8544203629
Circumradius: R = 1575.83658339

Vertex coordinates: A[2994; 0] B[0; 0] C[1573.979921772; 2066.143305912]
Centroid: CG[1522.665973924; 688.7144353039]
Coordinates of the circumscribed circle: U[1497; 492.1898556771]
Coordinates of the inscribed circle: I[1542.152196364; 763.8544203629]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 124.5° = 124°30' = 0.96986577349 rad
∠ B' = β' = 127.3° = 127°18' = 0.92197885158 rad
∠ C' = γ' = 108.2° = 108°12' = 1.25331464029 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 55° 30' ; ; beta = 52° 42' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 55° 30' - 52° 42' = 71° 48' ; ;

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

c = 2994 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 2994 * fraction{ sin(55° 30') }{ sin (71° 48') } = 2597.38 ; ;

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 = 2994 * fraction{ sin(52° 42') }{ sin (71° 48') } = 2507.07 ; ;


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 = 2597.38 ; ; b = 2507.07 ; ; c = 2994 ; ;

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

p = a+b+c = 2597.38+2507.07+2994 = 8098.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8098.45 }{ 2 } = 4049.22 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4049.22 * (4049.22-2597.38)(4049.22-2507.07)(4049.22-2994) } ; ; T = sqrt{ 9.567 * 10**{ 12 } } = 3093016.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3093016.16 }{ 2597.38 } = 2381.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3093016.16 }{ 2507.07 } = 2467.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3093016.16 }{ 2994 } = 2066.14 ; ;

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{ 2597.38**2-2507.07**2-2994**2 }{ 2 * 2507.07 * 2994 } ) = 55° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2507.07**2-2597.38**2-2994**2 }{ 2 * 2597.38 * 2994 } ) = 52° 42' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2994**2-2597.38**2-2507.07**2 }{ 2 * 2507.07 * 2597.38 } ) = 71° 48' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3093016.16 }{ 4049.22 } = 763.85 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2597.38 }{ 2 * sin 55° 30' } = 1575.84 ; ;




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