Triangle calculator ASA

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

Sides: a = 550   b = 952.6287944163   c = 1100

Area: T = 261972.6854645
Perimeter: p = 2602.628794416
Semiperimeter: s = 1301.314397208

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 952.6287944163
Height: hb = 550
Height: hc = 476.3143972081

Median: ma = 991.5276600753
Median: mb = 727.5821610543
Median: mc = 550

Inradius: r = 201.3143972081
Circumradius: R = 550

Vertex coordinates: A[1100; 0] B[0; 0] C[275; 476.3143972081]
Centroid: CG[458.3333333333; 158.7711324027]
Coordinates of the circumscribed circle: U[550; 0]
Coordinates of the inscribed circle: I[348.6866027919; 201.3143972081]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 30° ; ; beta = 60° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 30° - 60° = 90° ; ;

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

c = 1100 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 1100 * fraction{ sin(30° ) }{ sin (90° ) } = 550 ; ;

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 = 1100 * fraction{ sin(60° ) }{ sin (90° ) } = 952.63 ; ;


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 = 550 ; ; b = 952.63 ; ; c = 1100 ; ;

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

p = a+b+c = 550+952.63+1100 = 2602.63 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2602.63 }{ 2 } = 1301.31 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1301.31 * (1301.31-550)(1301.31-952.63)(1301.31-1100) } ; ; T = sqrt{ 68629687500 } = 261972.68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 261972.68 }{ 550 } = 952.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 261972.68 }{ 952.63 } = 550 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 261972.68 }{ 1100 } = 476.31 ; ;

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{ 550**2-952.63**2-1100**2 }{ 2 * 952.63 * 1100 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 952.63**2-550**2-1100**2 }{ 2 * 550 * 1100 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1100**2-550**2-952.63**2 }{ 2 * 952.63 * 550 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 261972.68 }{ 1301.31 } = 201.31 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 550 }{ 2 * sin 30° } = 550 ; ;




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