Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 96   b = 48   c = 83.13884387633

Area: T = 1995.323253032
Perimeter: p = 227.1388438763
Semiperimeter: s = 113.5699219382

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 41.56992193817
Height: hb = 83.13884387633
Height: hc = 48

Median: ma = 48
Median: mb = 86.53332306111
Median: mc = 63.49880314656

Inradius: r = 17.56992193817
Circumradius: R = 48

Vertex coordinates: A[83.13884387633; 0] B[0; 0] C[83.13884387633; 48]
Centroid: CG[55.42656258422; 16]
Coordinates of the circumscribed circle: U[41.56992193817; 24]
Coordinates of the inscribed circle: I[65.56992193817; 17.56992193817]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 30° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 30° = 60° ; ;

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

a = 96 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 96 * fraction{ sin 30° }{ sin 90° } = 48 ; ;

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 = 96 * fraction{ sin 60° }{ sin 90° } = 83.14 ; ;
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 = 96 ; ; b = 48 ; ; c = 83.14 ; ;

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

p = a+b+c = 96+48+83.14 = 227.14 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 227.14 }{ 2 } = 113.57 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 113.57 * (113.57-96)(113.57-48)(113.57-83.14) } ; ; T = sqrt{ 3981312 } = 1995.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1995.32 }{ 96 } = 41.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1995.32 }{ 48 } = 83.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1995.32 }{ 83.14 } = 48 ; ;

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**2+83.14**2-96**2 }{ 2 * 48 * 83.14 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 96**2+83.14**2-48**2 }{ 2 * 96 * 83.14 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 30° = 60° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1995.32 }{ 113.57 } = 17.57 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 96 }{ 2 * sin 90° } = 48 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 83.14**2 - 96**2 } }{ 2 } = 48 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 83.14**2+2 * 96**2 - 48**2 } }{ 2 } = 86.533 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 48**2+2 * 96**2 - 83.14**2 } }{ 2 } = 63.498 ; ;
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