Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 14.6   b = 23.51550901461   c = 32.99220856462

Area: T = 151.5676923355
Perimeter: p = 71.10771757923
Semiperimeter: s = 35.55435878962

Angle ∠ A = α = 23° = 0.4011425728 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 118° = 2.05994885174 rad

Height: ha = 20.76325922404
Height: hb = 12.89110348557
Height: hc = 9.18880777093

Median: ma = 27.70325015104
Median: mb = 22.64402074084
Median: mc = 10.53328202999

Inradius: r = 4.26330556386
Circumradius: R = 18.68329240563

Vertex coordinates: A[32.99220856462; 0] B[0; 0] C[11.34663310373; 9.18880777093]
Centroid: CG[14.77994722278; 3.06326925698]
Coordinates of the circumscribed circle: U[16.49660428231; -8.77111015541]
Coordinates of the inscribed circle: I[12.038849775; 4.26330556386]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157° = 0.4011425728 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 62° = 2.05994885174 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 23° ; ; beta = 39° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 23° - 39° = 118° ; ;

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

a = 14.6 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 14.6 * fraction{ sin 39° }{ sin 23° } = 23.52 ; ;

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 = 14.6 * fraction{ sin 118° }{ sin 23° } = 32.99 ; ;


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 = 14.6 ; ; b = 23.52 ; ; c = 32.99 ; ;

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

p = a+b+c = 14.6+23.52+32.99 = 71.11 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71.11 }{ 2 } = 35.55 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.55 * (35.55-14.6)(35.55-23.52)(35.55-32.99) } ; ; T = sqrt{ 22972.53 } = 151.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 151.57 }{ 14.6 } = 20.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 151.57 }{ 23.52 } = 12.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 151.57 }{ 32.99 } = 9.19 ; ;

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{ 23.52**2+32.99**2-14.6**2 }{ 2 * 23.52 * 32.99 } ) = 23° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.6**2+32.99**2-23.52**2 }{ 2 * 14.6 * 32.99 } ) = 39° ; ; gamma = 180° - alpha - beta = 180° - 23° - 39° = 118° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 151.57 }{ 35.55 } = 4.26 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 14.6 }{ 2 * sin 23° } = 18.68 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.52**2+2 * 32.99**2 - 14.6**2 } }{ 2 } = 27.703 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.99**2+2 * 14.6**2 - 23.52**2 } }{ 2 } = 22.64 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.52**2+2 * 14.6**2 - 32.99**2 } }{ 2 } = 10.533 ; ;
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