Triangle calculator ASA

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

Sides: a = 136.808805733   b = 69.45992710668   c = 200

Area: T = 2375.647698456
Perimeter: p = 406.2677328397
Semiperimeter: s = 203.1343664199

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 10° = 0.17545329252 rad
Angle ∠ C = γ = 150° = 2.6187993878 rad

Height: ha = 34.73296355334
Height: hb = 68.40440286651
Height: hc = 23.75664698456

Median: ma = 133.1666001783
Median: mb = 167.7865800028
Median: mc = 42.07875170822

Inradius: r = 11.69549940027
Circumradius: R = 200

Vertex coordinates: A[200; 0] B[0; 0] C[134.7329635533; 23.75664698456]
Centroid: CG[111.5776545178; 7.91988232819]
Coordinates of the circumscribed circle: U[100; -173.2055080757]
Coordinates of the inscribed circle: I[133.6744393132; 11.69549940027]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 170° = 0.17545329252 rad
∠ C' = γ' = 30° = 2.6187993878 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 20° ; ; beta = 10° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 20° - 10° = 150° ; ;

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

c = 200 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 200 * fraction{ sin(20° ) }{ sin (150° ) } = 136.81 ; ;

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 = 200 * fraction{ sin(10° ) }{ sin (150° ) } = 69.46 ; ;


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 = 136.81 ; ; b = 69.46 ; ; c = 200 ; ;

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

p = a+b+c = 136.81+69.46+200 = 406.27 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 406.27 }{ 2 } = 203.13 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 203.13 * (203.13-136.81)(203.13-69.46)(203.13-200) } ; ; T = sqrt{ 5643698.6 } = 2375.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2375.65 }{ 136.81 } = 34.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2375.65 }{ 69.46 } = 68.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2375.65 }{ 200 } = 23.76 ; ;

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{ 136.81**2-69.46**2-200**2 }{ 2 * 69.46 * 200 } ) = 20° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 69.46**2-136.81**2-200**2 }{ 2 * 136.81 * 200 } ) = 10° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200**2-136.81**2-69.46**2 }{ 2 * 69.46 * 136.81 } ) = 150° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2375.65 }{ 203.13 } = 11.69 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 136.81 }{ 2 * sin 20° } = 200 ; ;




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