Triangle calculator ASA

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


Acute scalene triangle.

Sides: a = 20.06106404638   b = 21.31768537518   c = 12.5

Area: T = 122.6399170868
Perimeter: p = 53.87774942156
Semiperimeter: s = 26.93987471078

Angle ∠ A = α = 67° = 1.16993705988 rad
Angle ∠ B = β = 78° = 1.36113568166 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: ha = 12.22768450092
Height: hb = 11.50663106682
Height: hc = 19.62222673389

Median: ma = 14.30881027038
Median: mb = 12.87439110002
Median: mc = 19.73221127821

Inradius: r = 4.5532519476
Circumradius: R = 10.89765424726

Vertex coordinates: A[12.5; 0] B[0; 0] C[4.17108416777; 19.62222673389]
Centroid: CG[5.55769472259; 6.54107557796]
Coordinates of the circumscribed circle: U[6.25; 8.92659250421]
Coordinates of the inscribed circle: I[5.6221893356; 4.5532519476]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113° = 1.16993705988 rad
∠ B' = β' = 102° = 1.36113568166 rad
∠ C' = γ' = 145° = 0.61108652382 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 67° ; ; beta = 78° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 67° - 78° = 35° ; ;

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

c = 12.5 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 12.5 * fraction{ sin(67° ) }{ sin (35° ) } = 20.06 ; ;

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 = 12.5 * fraction{ sin(78° ) }{ sin (35° ) } = 21.32 ; ;


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 = 20.06 ; ; b = 21.32 ; ; c = 12.5 ; ;

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

p = a+b+c = 20.06+21.32+12.5 = 53.88 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53.88 }{ 2 } = 26.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.94 * (26.94-20.06)(26.94-21.32)(26.94-12.5) } ; ; T = sqrt{ 15040.37 } = 122.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 122.64 }{ 20.06 } = 12.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 122.64 }{ 21.32 } = 11.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 122.64 }{ 12.5 } = 19.62 ; ;

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{ 20.06**2-21.32**2-12.5**2 }{ 2 * 21.32 * 12.5 } ) = 67° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.32**2-20.06**2-12.5**2 }{ 2 * 20.06 * 12.5 } ) = 78° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.5**2-20.06**2-21.32**2 }{ 2 * 21.32 * 20.06 } ) = 35° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 122.64 }{ 26.94 } = 4.55 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.06 }{ 2 * sin 67° } = 10.9 ; ;




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