Triangle calculator ASA

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


Acute scalene triangle.

Sides: a = 15.09110689502   b = 15.80773075776   c = 6.2

Area: T = 46.60442929179
Perimeter: p = 37.09883765278
Semiperimeter: s = 18.54991882639

Angle ∠ A = α = 72° = 1.25766370614 rad
Angle ∠ B = β = 85° = 1.48435298642 rad
Angle ∠ C = γ = 23° = 0.4011425728 rad

Height: ha = 6.17664071282
Height: hb = 5.8976550401
Height: hc = 15.03436428767

Median: ma = 9.33991860411
Median: mb = 8.40437157149
Median: mc = 15.13992096047

Inradius: r = 2.51224707483
Circumradius: R = 7.93438444623

Vertex coordinates: A[6.2; 0] B[0; 0] C[1.31552733232; 15.03436428767]
Centroid: CG[2.50550911077; 5.01112142922]
Coordinates of the circumscribed circle: U[3.1; 7.30331423341]
Coordinates of the inscribed circle: I[2.74218806863; 2.51224707483]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108° = 1.25766370614 rad
∠ B' = β' = 95° = 1.48435298642 rad
∠ C' = γ' = 157° = 0.4011425728 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 72° ; ; beta = 85° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 72° - 85° = 23° ; ;

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

c = 6.2 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 6.2 * fraction{ sin(72° ) }{ sin (23° ) } = 15.09 ; ;

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 = 6.2 * fraction{ sin(85° ) }{ sin (23° ) } = 15.81 ; ;


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 = 15.09 ; ; b = 15.81 ; ; c = 6.2 ; ;

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

p = a+b+c = 15.09+15.81+6.2 = 37.1 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.1 }{ 2 } = 18.55 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.55 * (18.55-15.09)(18.55-15.81)(18.55-6.2) } ; ; T = sqrt{ 2171.96 } = 46.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46.6 }{ 15.09 } = 6.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46.6 }{ 15.81 } = 5.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46.6 }{ 6.2 } = 15.03 ; ;

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{ 15.09**2-15.81**2-6.2**2 }{ 2 * 15.81 * 6.2 } ) = 72° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15.81**2-15.09**2-6.2**2 }{ 2 * 15.09 * 6.2 } ) = 85° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.2**2-15.09**2-15.81**2 }{ 2 * 15.81 * 15.09 } ) = 23° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46.6 }{ 18.55 } = 2.51 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15.09 }{ 2 * sin 72° } = 7.93 ; ;




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