Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 0.81223318287   b = 0.469   c = 0.938

Area: T = 0.19904918138
Perimeter: p = 2.21993318287
Semiperimeter: s = 1.11096659144

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

Height: ha = 0.469
Height: hb = 0.81223318287
Height: hc = 0.40661659144

Median: ma = 0.62204286824
Median: mb = 0.84655017741
Median: mc = 0.469

Inradius: r = 0.17216659144
Circumradius: R = 0.469

Vertex coordinates: A[0.938; 0] B[0; 0] C[0.70435; 0.40661659144]
Centroid: CG[0.54771666667; 0.13553886381]
Coordinates of the circumscribed circle: U[0.469; -0]
Coordinates of the inscribed circle: I[0.64106659144; 0.17216659144]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 150° = 0.52435987756 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 = 60° ; ; beta = 30° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 60° - 30° = 90° ; ;

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

c = 0.94 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 0.94 * fraction{ sin(60° ) }{ sin (90° ) } = 0.81 ; ;

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 = 0.94 * fraction{ sin(30° ) }{ sin (90° ) } = 0.47 ; ;


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 = 0.81 ; ; b = 0.47 ; ; c = 0.94 ; ;

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

p = a+b+c = 0.81+0.47+0.94 = 2.22 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.22 }{ 2 } = 1.11 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.11 * (1.11-0.81)(1.11-0.47)(1.11-0.94) } ; ; T = sqrt{ 0.04 } = 0.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.19 }{ 0.81 } = 0.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.19 }{ 0.47 } = 0.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.19 }{ 0.94 } = 0.41 ; ;

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{ 0.81**2-0.47**2-0.94**2 }{ 2 * 0.47 * 0.94 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.47**2-0.81**2-0.94**2 }{ 2 * 0.81 * 0.94 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.94**2-0.81**2-0.47**2 }{ 2 * 0.47 * 0.81 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.19 }{ 1.11 } = 0.17 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 0.81 }{ 2 * sin 60° } = 0.47 ; ;




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