Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 47   b = 16.07549467363   c = 44.16655531769

Area: T = 354.9799457449
Perimeter: p = 107.2440499913
Semiperimeter: s = 53.62202499566

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 70° = 1.22217304764 rad

Height: ha = 15.10655088276
Height: hb = 44.16655531769
Height: hc = 16.07549467363

Median: ma = 23.5
Median: mb = 44.89109463653
Median: mc = 27.31439695839

Inradius: r = 6.62202499566
Circumradius: R = 23.5

Vertex coordinates: A[44.16655531769; 0] B[0; 0] C[44.16655531769; 16.07549467363]
Centroid: CG[29.4443702118; 5.35883155788]
Coordinates of the circumscribed circle: U[22.08327765885; 8.03774733682]
Coordinates of the inscribed circle: I[37.54553032203; 6.62202499566]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 110° = 1.22217304764 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 90° ; ; beta = 20° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 20° = 70° ; ;

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

a = 47 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 47 * fraction{ sin 20° }{ sin 90° } = 16.07 ; ;

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 = 47 * fraction{ sin 70° }{ sin 90° } = 44.17 ; ;


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 = 47 ; ; b = 16.07 ; ; c = 44.17 ; ;

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

p = a+b+c = 47+16.07+44.17 = 107.24 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 107.24 }{ 2 } = 53.62 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 53.62 * (53.62-47)(53.62-16.07)(53.62-44.17) } ; ; T = sqrt{ 126010.42 } = 354.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 354.98 }{ 47 } = 15.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 354.98 }{ 16.07 } = 44.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 354.98 }{ 44.17 } = 16.07 ; ;

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{ 16.07**2+44.17**2-47**2 }{ 2 * 16.07 * 44.17 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 47**2+44.17**2-16.07**2 }{ 2 * 47 * 44.17 } ) = 20° ; ; gamma = 180° - alpha - beta = 180° - 90° - 20° = 70° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 354.98 }{ 53.62 } = 6.62 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 47 }{ 2 * sin 90° } = 23.5 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.07**2+2 * 44.17**2 - 47**2 } }{ 2 } = 23.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44.17**2+2 * 47**2 - 16.07**2 } }{ 2 } = 44.891 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.07**2+2 * 47**2 - 44.17**2 } }{ 2 } = 27.314 ; ;
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