Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 2.5   b = 3.09550895413   c = 4.51215587435

Area: T = 3.76998110606
Perimeter: p = 10.10766482848
Semiperimeter: s = 5.05333241424

Angle ∠ A = α = 32° = 0.55985053606 rad
Angle ∠ B = β = 41° = 0.71655849933 rad
Angle ∠ C = γ = 107° = 1.86875022996 rad

Height: ha = 2.96598488485
Height: hb = 2.39107618899
Height: hc = 1.64401475725

Median: ma = 3.66111979983
Median: mb = 3.30326029629
Median: mc = 1.68111451634

Inradius: r = 0.73221539162
Circumradius: R = 2.35988498935

Vertex coordinates: A[4.51215587435; 0] B[0; 0] C[1.88767739506; 1.64401475725]
Centroid: CG[2.13327775647; 0.54767158575]
Coordinates of the circumscribed circle: U[2.25657793718; -0.69896609645]
Coordinates of the inscribed circle: I[1.95882346011; 0.73221539162]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148° = 0.55985053606 rad
∠ B' = β' = 139° = 0.71655849933 rad
∠ C' = γ' = 73° = 1.86875022996 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 32° ; ; beta = 41° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 32° - 41° = 107° ; ;

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

a = 2.5 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 2.5 * fraction{ sin(41° ) }{ sin (32° ) } = 3.1 ; ;

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 = 2.5 * fraction{ sin(107° ) }{ sin (32° ) } = 4.51 ; ;


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 = 2.5 ; ; b = 3.1 ; ; c = 4.51 ; ;

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

p = a+b+c = 2.5+3.1+4.51 = 10.11 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.11 }{ 2 } = 5.05 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.05 * (5.05-2.5)(5.05-3.1)(5.05-4.51) } ; ; T = sqrt{ 13.69 } = 3.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.7 }{ 2.5 } = 2.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.7 }{ 3.1 } = 2.39 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.7 }{ 4.51 } = 1.64 ; ;

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{ 2.5**2-3.1**2-4.51**2 }{ 2 * 3.1 * 4.51 } ) = 32° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.1**2-2.5**2-4.51**2 }{ 2 * 2.5 * 4.51 } ) = 41° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.51**2-2.5**2-3.1**2 }{ 2 * 3.1 * 2.5 } ) = 107° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.7 }{ 5.05 } = 0.73 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.5 }{ 2 * sin 32° } = 2.36 ; ;




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