Triangle calculator ASA

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

Sides: a = 529.4565621705   b = 869.7265965667   c = 690

Area: T = 182662.1899488
Perimeter: p = 2089.182158737
Semiperimeter: s = 1044.591079369

Angle ∠ A = α = 37.5° = 37°30' = 0.65444984695 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 52.5° = 52°30' = 0.91662978573 rad

Height: ha = 690
Height: hb = 420.0455386016
Height: hc = 529.4565621705

Median: ma = 739.0440468336
Median: mb = 434.8632982833
Median: mc = 631.9440072598

Inradius: r = 174.8654828019
Circumradius: R = 434.8632982833

Vertex coordinates: A[690; 0] B[0; 0] C[-0; 529.4565621705]
Centroid: CG[230; 176.4855207235]
Coordinates of the circumscribed circle: U[345; 264.7287810853]
Coordinates of the inscribed circle: I[174.8654828019; 174.8654828019]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.5° = 142°30' = 0.65444984695 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 127.5° = 127°30' = 0.91662978573 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 37° 30' ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 37° 30' - 90° = 52° 30' ; ;

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

c = 690 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 690 * fraction{ sin(37° 30') }{ sin (52° 30') } = 529.46 ; ;

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 = 690 * fraction{ sin(90° ) }{ sin (52° 30') } = 869.73 ; ;


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 = 529.46 ; ; b = 869.73 ; ; c = 690 ; ;

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

p = a+b+c = 529.46+869.73+690 = 2089.18 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2089.18 }{ 2 } = 1044.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1044.59 * (1044.59-529.46)(1044.59-869.73)(1044.59-690) } ; ; T = sqrt{ 33365475468.7 } = 182662.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 182662.19 }{ 529.46 } = 690 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 182662.19 }{ 869.73 } = 420.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 182662.19 }{ 690 } = 529.46 ; ;

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{ 529.46**2-869.73**2-690**2 }{ 2 * 869.73 * 690 } ) = 37° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 869.73**2-529.46**2-690**2 }{ 2 * 529.46 * 690 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 690**2-529.46**2-869.73**2 }{ 2 * 869.73 * 529.46 } ) = 52° 30' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 182662.19 }{ 1044.59 } = 174.86 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 529.46 }{ 2 * sin 37° 30' } = 434.86 ; ;




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