Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 537.9533452899   b = 390.2688459837   c = 675

Area: T = 104915.4220208
Perimeter: p = 1603.222191274
Semiperimeter: s = 801.6110956368

Angle ∠ A = α = 52.8° = 52°48' = 0.92215338451 rad
Angle ∠ B = β = 35.3° = 35°18' = 0.6166101226 rad
Angle ∠ C = γ = 91.9° = 91°54' = 1.60439575826 rad

Height: ha = 390.0543896459
Height: hb = 537.6587694666
Height: hc = 310.8610504321

Median: ma = 481.2687863045
Median: mb = 578.3011038437
Median: mc = 327.0255142939

Inradius: r = 130.8810721346
Circumradius: R = 337.6865654189

Vertex coordinates: A[675; 0] B[0; 0] C[439.0444034624; 310.8610504321]
Centroid: CG[371.3488011541; 103.6220168107]
Coordinates of the circumscribed circle: U[337.5; -11.19660281039]
Coordinates of the inscribed circle: I[411.3422496531; 130.8810721346]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.2° = 127°12' = 0.92215338451 rad
∠ B' = β' = 144.7° = 144°42' = 0.6166101226 rad
∠ C' = γ' = 88.1° = 88°6' = 1.60439575826 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 52° 48' ; ; beta = 35° 18' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 52° 48' - 35° 18' = 91° 54' ; ;

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

c = 675 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 675 * fraction{ sin(52° 48') }{ sin (91° 54') } = 537.95 ; ;

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 = 675 * fraction{ sin(35° 18') }{ sin (91° 54') } = 390.27 ; ;


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 = 537.95 ; ; b = 390.27 ; ; c = 675 ; ;

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

p = a+b+c = 537.95+390.27+675 = 1603.22 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1603.22 }{ 2 } = 801.61 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 801.61 * (801.61-537.95)(801.61-390.27)(801.61-675) } ; ; T = sqrt{ 11007245397.5 } = 104915.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104915.42 }{ 537.95 } = 390.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104915.42 }{ 390.27 } = 537.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104915.42 }{ 675 } = 310.86 ; ;

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{ 537.95**2-390.27**2-675**2 }{ 2 * 390.27 * 675 } ) = 52° 48' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 390.27**2-537.95**2-675**2 }{ 2 * 537.95 * 675 } ) = 35° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 675**2-537.95**2-390.27**2 }{ 2 * 390.27 * 537.95 } ) = 91° 54' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104915.42 }{ 801.61 } = 130.88 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 537.95 }{ 2 * sin 52° 48' } = 337.69 ; ;




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