Triangle calculator ASA

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


Acute scalene triangle.

Sides: a = 11.08223604879   b = 12.9410902488   c = 8.4

Area: T = 46.09329324134
Perimeter: p = 32.42332629759
Semiperimeter: s = 16.21216314879

Angle ∠ A = α = 58° = 1.01222909662 rad
Angle ∠ B = β = 82° = 1.43111699866 rad
Angle ∠ C = γ = 40° = 0.69881317008 rad

Height: ha = 8.31882517774
Height: hb = 7.12436040077
Height: hc = 10.97545077175

Median: ma = 9.39772762067
Median: mb = 7.40442297163
Median: mc = 11.29217153521

Inradius: r = 2.84332013427
Circumradius: R = 6.53440400728

Vertex coordinates: A[8.4; 0] B[0; 0] C[1.54223664751; 10.97545077175]
Centroid: CG[3.31441221584; 3.65881692392]
Coordinates of the circumscribed circle: U[4.2; 5.00553650889]
Coordinates of the inscribed circle: I[3.2710729; 2.84332013427]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122° = 1.01222909662 rad
∠ B' = β' = 98° = 1.43111699866 rad
∠ C' = γ' = 140° = 0.69881317008 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 58° ; ; beta = 82° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 58° - 82° = 40° ; ;

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

c = 8.4 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 8.4 * fraction{ sin(58° ) }{ sin (40° ) } = 11.08 ; ;

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 = 8.4 * fraction{ sin(82° ) }{ sin (40° ) } = 12.94 ; ;


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 = 11.08 ; ; b = 12.94 ; ; c = 8.4 ; ;

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

p = a+b+c = 11.08+12.94+8.4 = 32.42 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.42 }{ 2 } = 16.21 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.21 * (16.21-11.08)(16.21-12.94)(16.21-8.4) } ; ; T = sqrt{ 2124.56 } = 46.09 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46.09 }{ 11.08 } = 8.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46.09 }{ 12.94 } = 7.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46.09 }{ 8.4 } = 10.97 ; ;

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{ 11.08**2-12.94**2-8.4**2 }{ 2 * 12.94 * 8.4 } ) = 58° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.94**2-11.08**2-8.4**2 }{ 2 * 11.08 * 8.4 } ) = 82° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.4**2-11.08**2-12.94**2 }{ 2 * 12.94 * 11.08 } ) = 40° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46.09 }{ 16.21 } = 2.84 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.08 }{ 2 * sin 58° } = 6.53 ; ;




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