Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 2.18875   b = 3.63548378087   c = 2.90329105473

Area: T = 3.17550584111
Perimeter: p = 8.7255248356
Semiperimeter: s = 4.3632624178

Angle ∠ A = α = 37° = 0.64657718232 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 53° = 0.92550245036 rad

Height: ha = 2.90329105473
Height: hb = 1.74770151782
Height: hc = 2.18875

Median: ma = 3.10221248698
Median: mb = 1.81774189044
Median: mc = 2.62552387818

Inradius: r = 0.72877863693
Circumradius: R = 1.81774189044

Vertex coordinates: A[2.90329105473; 0] B[0; 0] C[-0; 2.18875]
Centroid: CG[0.96876368491; 0.72991666667]
Coordinates of the circumscribed circle: U[1.45114552736; 1.094375]
Coordinates of the inscribed circle: I[0.72877863693; 0.72877863693]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143° = 0.64657718232 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 127° = 0.92550245036 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 37° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 37° - 90° = 53° ; ;

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

a = 2.19 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 2.19 * fraction{ sin 90° }{ sin 37° } = 3.63 ; ;

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 = 2.19 * fraction{ sin 53° }{ sin 37° } = 2.9 ; ;
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 = 2.19 ; ; b = 3.63 ; ; c = 2.9 ; ;

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

p = a+b+c = 2.19+3.63+2.9 = 8.73 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8.73 }{ 2 } = 4.36 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4.36 * (4.36-2.19)(4.36-3.63)(4.36-2.9) } ; ; T = sqrt{ 10.08 } = 3.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.18 }{ 2.19 } = 2.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.18 }{ 3.63 } = 1.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.18 }{ 2.9 } = 2.19 ; ;

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{ 3.63**2+2.9**2-2.19**2 }{ 2 * 3.63 * 2.9 } ) = 37° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.19**2+2.9**2-3.63**2 }{ 2 * 2.19 * 2.9 } ) = 90° ; ;
 gamma = 180° - alpha - beta = 180° - 37° - 90° = 53° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.18 }{ 4.36 } = 0.73 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 2.19 }{ 2 * sin 37° } = 1.82 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.63**2+2 * 2.9**2 - 2.19**2 } }{ 2 } = 3.102 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.9**2+2 * 2.19**2 - 3.63**2 } }{ 2 } = 1.817 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.63**2+2 * 2.19**2 - 2.9**2 } }{ 2 } = 2.625 ; ;
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