Triangle calculator ASA

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

Sides: a = 688.3177307256   b = 547.7321929497   c = 500

Area: T = 135600.3660003
Perimeter: p = 1736.049923675
Semiperimeter: s = 868.0254618376

Angle ∠ A = α = 82° = 1.43111699866 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 46° = 0.80328514559 rad

Height: ha = 394.0055376803
Height: hb = 495.1344034371
Height: hc = 542.4011440011

Median: ma = 395.6776578064
Median: mb = 535.6199072743
Median: mc = 569.5577276337

Inradius: r = 156.2177182246
Circumradius: R = 347.5410897754

Vertex coordinates: A[500; 0] B[0; 0] C[423.7770448877; 542.4011440011]
Centroid: CG[307.9233482959; 180.8800480004]
Coordinates of the circumscribed circle: U[250; 241.4222193702]
Coordinates of the inscribed circle: I[320.2932688879; 156.2177182246]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98° = 1.43111699866 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 134° = 0.80328514559 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 82° ; ; beta = 52° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 82° - 52° = 46° ; ;

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

c = 500 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 500 * fraction{ sin(82° ) }{ sin (46° ) } = 688.32 ; ;

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 = 500 * fraction{ sin(52° ) }{ sin (46° ) } = 547.73 ; ;


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 = 688.32 ; ; b = 547.73 ; ; c = 500 ; ;

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

p = a+b+c = 688.32+547.73+500 = 1736.05 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1736.05 }{ 2 } = 868.02 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 868.02 * (868.02-688.32)(868.02-547.73)(868.02-500) } ; ; T = sqrt{ 18387457632.9 } = 135600.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 135600.36 }{ 688.32 } = 394.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 135600.36 }{ 547.73 } = 495.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 135600.36 }{ 500 } = 542.4 ; ;

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{ 688.32**2-547.73**2-500**2 }{ 2 * 547.73 * 500 } ) = 82° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 547.73**2-688.32**2-500**2 }{ 2 * 688.32 * 500 } ) = 52° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 500**2-688.32**2-547.73**2 }{ 2 * 547.73 * 688.32 } ) = 46° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 135600.36 }{ 868.02 } = 156.22 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 688.32 }{ 2 * sin 82° } = 347.54 ; ;




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