Triangle calculator AAS

Please enter two angles and one opposite side
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°


Acute scalene triangle.

Sides: a = 31   b = 48.40217820968   c = 46.68658828163

Area: T = 695.5998938832
Perimeter: p = 126.0887664913
Semiperimeter: s = 63.04438324565

Angle ∠ A = α = 38° = 0.66332251158 rad
Angle ∠ B = β = 74° = 1.29215436465 rad
Angle ∠ C = γ = 68° = 1.18768238914 rad

Height: ha = 44.87773508924
Height: hb = 28.74326994916
Height: hc = 29.79991125741

Median: ma = 44.95444445215
Median: mb = 31.37883794934
Median: mc = 33.27112088973

Inradius: r = 11.03435763504
Circumradius: R = 25.1766173305

Vertex coordinates: A[46.68658828163; 0] B[0; 0] C[8.54547580303; 29.79991125741]
Centroid: CG[18.41102136155; 9.93330375247]
Coordinates of the circumscribed circle: U[23.34329414082; 9.4311160517]
Coordinates of the inscribed circle: I[14.64220503598; 11.03435763504]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142° = 0.66332251158 rad
∠ B' = β' = 106° = 1.29215436465 rad
∠ C' = γ' = 112° = 1.18768238914 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 38° ; ; beta = 74° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 38° - 74° = 68° ; ;

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

a = 31 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 31 * fraction{ sin 74° }{ sin 38° } = 48.4 ; ;

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 = 31 * fraction{ sin 68° }{ sin 38° } = 46.69 ; ;


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 = 31 ; ; b = 48.4 ; ; c = 46.69 ; ;

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

p = a+b+c = 31+48.4+46.69 = 126.09 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 126.09 }{ 2 } = 63.04 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.04 * (63.04-31)(63.04-48.4)(63.04-46.69) } ; ; T = sqrt{ 483857.88 } = 695.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 695.6 }{ 31 } = 44.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 695.6 }{ 48.4 } = 28.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 695.6 }{ 46.69 } = 29.8 ; ;

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{ 48.4**2+46.69**2-31**2 }{ 2 * 48.4 * 46.69 } ) = 38° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31**2+46.69**2-48.4**2 }{ 2 * 31 * 46.69 } ) = 74° ; ; gamma = 180° - alpha - beta = 180° - 38° - 74° = 68° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 695.6 }{ 63.04 } = 11.03 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 31 }{ 2 * sin 38° } = 25.18 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.4**2+2 * 46.69**2 - 31**2 } }{ 2 } = 44.954 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 46.69**2+2 * 31**2 - 48.4**2 } }{ 2 } = 31.378 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48.4**2+2 * 31**2 - 46.69**2 } }{ 2 } = 33.271 ; ;
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