Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 90   b = 155.8854572681   c = 180

Area: T = 7014.806577065
Perimeter: p = 425.8854572681
Semiperimeter: s = 212.9422286341

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

Height: ha = 155.8854572681
Height: hb = 90
Height: hc = 77.94222863406

Median: ma = 162.2549807396
Median: mb = 119.0598808998
Median: mc = 90

Inradius: r = 32.94222863406
Circumradius: R = 90

Vertex coordinates: A[180; 0] B[0; 0] C[45; 77.94222863406]
Centroid: CG[75; 25.98107621135]
Coordinates of the circumscribed circle: U[90; -0]
Coordinates of the inscribed circle: I[57.05877136594; 32.94222863406]

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

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

a = 90 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 90 * fraction{ sin 60° }{ sin 30° } = 155.88 ; ;

3. By using the law of sines, we calculate last unknown side c

fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 90 * fraction{ sin 90° }{ sin 30° } = 180 ; ;


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 = 90 ; ; b = 155.88 ; ; c = 180 ; ;

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

p = a+b+c = 90+155.88+180 = 425.88 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 425.88 }{ 2 } = 212.94 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 212.94 * (212.94-90)(212.94-155.88)(212.94-180) } ; ; T = sqrt{ 49207500 } = 7014.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7014.81 }{ 90 } = 155.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7014.81 }{ 155.88 } = 90 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7014.81 }{ 180 } = 77.94 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 155.88**2+180**2-90**2 }{ 2 * 155.88 * 180 } ) = 30° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+180**2-155.88**2 }{ 2 * 90 * 180 } ) = 60° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 90**2+155.88**2-180**2 }{ 2 * 90 * 155.88 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7014.81 }{ 212.94 } = 32.94 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 30° } = 90 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 155.88**2+2 * 180**2 - 90**2 } }{ 2 } = 162.25 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 180**2+2 * 90**2 - 155.88**2 } }{ 2 } = 119.059 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 155.88**2+2 * 90**2 - 180**2 } }{ 2 } = 90 ; ;
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