Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 120   b = 89.06772638762   c = 136.4598965112

Area: T = 5262.848792029
Perimeter: p = 345.5266228989
Semiperimeter: s = 172.7633114494

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 87.71441320049
Height: hb = 118.1776930361
Height: hc = 77.13545131624

Median: ma = 98.37218116483
Median: mb = 120.529916745
Median: mc = 80.69221709791

Inradius: r = 30.46327983566
Circumradius: R = 69.28220323028

Vertex coordinates: A[136.4598965112; 0] B[0; 0] C[91.92553331743; 77.13545131624]
Centroid: CG[76.12880994289; 25.71215043875]
Coordinates of the circumscribed circle: U[68.22994825562; 12.03106986544]
Coordinates of the inscribed circle: I[83.69658506181; 30.46327983566]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 100° = 1.39662634016 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 60° ; ; beta = 40° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 60° - 40° = 80° ; ;

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

a = 120 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 120 * fraction{ sin 40° }{ sin 60° } = 89.07 ; ;

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 = 120 * fraction{ sin 80° }{ sin 60° } = 136.46 ; ;
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 = 120 ; ; b = 89.07 ; ; c = 136.46 ; ;

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

p = a+b+c = 120+89.07+136.46 = 345.53 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 345.53 }{ 2 } = 172.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 172.76 * (172.76-120)(172.76-89.07)(172.76-136.46) } ; ; T = sqrt{ 27697568.23 } = 5262.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5262.85 }{ 120 } = 87.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5262.85 }{ 89.07 } = 118.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5262.85 }{ 136.46 } = 77.13 ; ;

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{ 89.07**2+136.46**2-120**2 }{ 2 * 89.07 * 136.46 } ) = 60° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 120**2+136.46**2-89.07**2 }{ 2 * 120 * 136.46 } ) = 40° ; ;
 gamma = 180° - alpha - beta = 180° - 60° - 40° = 80° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5262.85 }{ 172.76 } = 30.46 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 120 }{ 2 * sin 60° } = 69.28 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 89.07**2+2 * 136.46**2 - 120**2 } }{ 2 } = 98.372 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 136.46**2+2 * 120**2 - 89.07**2 } }{ 2 } = 120.529 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 89.07**2+2 * 120**2 - 136.46**2 } }{ 2 } = 80.692 ; ;
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