Triangle calculator ASA

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


Obtuse scalene triangle.

Sides: a = 2.78993958927   b = 5.40766271022   c = 6.5

Area: T = 7.42660528804
Perimeter: p = 14.69660229948
Semiperimeter: s = 7.34880114974

Angle ∠ A = α = 25° = 0.4366332313 rad
Angle ∠ B = β = 55° = 0.96599310886 rad
Angle ∠ C = γ = 100° = 1.7455329252 rad

Height: ha = 5.32444882879
Height: hb = 2.74770187013
Height: hc = 2.28549393478

Median: ma = 5.81334005496
Median: mb = 4.20880233563
Median: mc = 2.81884522409

Inradius: r = 1.01106207486
Circumradius: R = 3.33001364886

Vertex coordinates: A[6.5; 0] B[0; 0] C[1.65999317557; 2.28549393478]
Centroid: CG[2.76999772519; 0.76216464493]
Coordinates of the circumscribed circle: U[3.25; -0.57330626873]
Coordinates of the inscribed circle: I[1.94113843952; 1.01106207486]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155° = 0.4366332313 rad
∠ B' = β' = 125° = 0.96599310886 rad
∠ C' = γ' = 80° = 1.7455329252 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 25° ; ; beta = 55° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 25° - 55° = 100° ; ;

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

c = 6.5 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 6.5 * fraction{ sin(25° ) }{ sin (100° ) } = 2.79 ; ;

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 = 6.5 * fraction{ sin(55° ) }{ sin (100° ) } = 5.41 ; ;


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 = 2.79 ; ; b = 5.41 ; ; c = 6.5 ; ;

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

p = a+b+c = 2.79+5.41+6.5 = 14.7 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.7 }{ 2 } = 7.35 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.35 * (7.35-2.79)(7.35-5.41)(7.35-6.5) } ; ; T = sqrt{ 55.15 } = 7.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.43 }{ 2.79 } = 5.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.43 }{ 5.41 } = 2.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.43 }{ 6.5 } = 2.28 ; ;

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{ 2.79**2-5.41**2-6.5**2 }{ 2 * 5.41 * 6.5 } ) = 25° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.41**2-2.79**2-6.5**2 }{ 2 * 2.79 * 6.5 } ) = 55° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.5**2-2.79**2-5.41**2 }{ 2 * 5.41 * 2.79 } ) = 100° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.43 }{ 7.35 } = 1.01 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.79 }{ 2 * sin 25° } = 3.3 ; ;




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