Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 70   b = 71.56438416406   c = 14.87989593169

Area: T = 520.7643576092
Perimeter: p = 156.4432800957
Semiperimeter: s = 78.22114004787

Angle ∠ A = α = 78° = 1.36113568166 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 12° = 0.20994395102 rad

Height: ha = 14.87989593169
Height: hb = 14.55438183572
Height: hc = 70

Median: ma = 38.03113479955
Median: mb = 35.78219208203
Median: mc = 70.39442175011

Inradius: r = 6.65875588382
Circumradius: R = 35.78219208203

Vertex coordinates: A[14.87989593169; 0] B[0; 0] C[0; 70]
Centroid: CG[4.96596531056; 23.33333333333]
Coordinates of the circumscribed circle: U[7.43994796585; 35]
Coordinates of the inscribed circle: I[6.65875588382; 6.65875588382]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102° = 1.36113568166 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 168° = 0.20994395102 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 78° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 78° - 90° = 12° ; ;

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

a = 70 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 70 * fraction{ sin(90° ) }{ sin (78° ) } = 71.56 ; ;

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 = 70 * fraction{ sin(12° ) }{ sin (78° ) } = 14.88 ; ;


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 = 70 ; ; b = 71.56 ; ; c = 14.88 ; ;

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

p = a+b+c = 70+71.56+14.88 = 156.44 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 156.44 }{ 2 } = 78.22 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 78.22 * (78.22-70)(78.22-71.56)(78.22-14.88) } ; ; T = sqrt{ 271194.7 } = 520.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 520.76 }{ 70 } = 14.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 520.76 }{ 71.56 } = 14.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 520.76 }{ 14.88 } = 70 ; ;

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{ 70**2-71.56**2-14.88**2 }{ 2 * 71.56 * 14.88 } ) = 78° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 71.56**2-70**2-14.88**2 }{ 2 * 70 * 14.88 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.88**2-70**2-71.56**2 }{ 2 * 71.56 * 70 } ) = 12° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 520.76 }{ 78.22 } = 6.66 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 70 }{ 2 * sin 78° } = 35.78 ; ;




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