Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 105   b = 85.77700522846   c = 127.226588761

Area: T = 4469.364360436
Perimeter: p = 317.9965939895
Semiperimeter: s = 158.9987969947

Angle ∠ A = α = 55° = 0.96599310886 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 83° = 1.44986232792 rad

Height: ha = 85.13107353211
Height: hb = 104.2177345922
Height: hc = 70.25987136677

Median: ma = 94.94884816811
Median: mb = 108.4743903644
Median: mc = 71.72326903769

Inradius: r = 28.11095639513
Circumradius: R = 64.091066591

Vertex coordinates: A[127.226588761; 0] B[0; 0] C[78.03302066751; 70.25987136677]
Centroid: CG[68.41986980951; 23.42195712226]
Coordinates of the circumscribed circle: U[63.61329438051; 7.81106873728]
Coordinates of the inscribed circle: I[73.22879176628; 28.11095639513]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125° = 0.96599310886 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 97° = 1.44986232792 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 55° ; ; beta = 42° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 55° - 42° = 83° ; ;

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

a = 105 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 105 * fraction{ sin(42° ) }{ sin (55° ) } = 85.77 ; ;

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 = 105 * fraction{ sin(83° ) }{ sin (55° ) } = 127.23 ; ;


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 = 105 ; ; b = 85.77 ; ; c = 127.23 ; ;

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

p = a+b+c = 105+85.77+127.23 = 318 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 318 }{ 2 } = 159 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 159 * (159-105)(159-85.77)(159-127.23) } ; ; T = sqrt{ 19975211.03 } = 4469.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4469.36 }{ 105 } = 85.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4469.36 }{ 85.77 } = 104.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4469.36 }{ 127.23 } = 70.26 ; ;

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{ 105**2-85.77**2-127.23**2 }{ 2 * 85.77 * 127.23 } ) = 55° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 85.77**2-105**2-127.23**2 }{ 2 * 105 * 127.23 } ) = 42° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 127.23**2-105**2-85.77**2 }{ 2 * 85.77 * 105 } ) = 83° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4469.36 }{ 159 } = 28.11 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 105 }{ 2 * sin 55° } = 64.09 ; ;




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