Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, angle α, angle β and angle γ.

Right scalene triangle.

Sides: a = 100   b = 93.96992620786   c = 34.20220143326

Area: T = 1606.969902422
Perimeter: p = 228.1711276411
Semiperimeter: s = 114.0865638206

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

Height: ha = 32.13993804843
Height: hb = 34.20220143326
Height: hc = 93.96992620786

Median: ma = 50
Median: mb = 58.1154828902
Median: mc = 95.5132651841

Inradius: r = 14.08656382056
Circumradius: R = 50

Vertex coordinates: A[34.20220143326; 0] B[0; 0] C[34.20220143326; 93.96992620786]
Centroid: CG[22.80113428884; 31.32330873595]
Coordinates of the circumscribed circle: U[17.10110071663; 46.98546310393]
Coordinates of the inscribed circle: I[20.1166376127; 14.08656382056]

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

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How did we calculate this triangle?

1. Input data entered: side a, angle α, angle β and angle γ.

a = 100 ; ; alpha = 90° ; ; beta = 70° ; ; gamma = 20° ; ;

2. From angle β, angle α and side a we calculate side b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 100 * fraction{ sin 70° }{ sin 90° } = 93.97 ; ;

3. From angle γ, angle α and side a we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 100 * fraction{ sin 20° }{ sin 90° } = 34.2 ; ;
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 = 100 ; ; b = 93.97 ; ; c = 34.2 ; ;

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

p = a+b+c = 100+93.97+34.2 = 228.17 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 228.17 }{ 2 } = 114.09 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 114.09 * (114.09-100)(114.09-93.97)(114.09-34.2) } ; ; T = sqrt{ 2582349.44 } = 1606.97 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1606.97 }{ 100 } = 32.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1606.97 }{ 93.97 } = 34.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1606.97 }{ 34.2 } = 93.97 ; ;

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{ 93.97**2+34.2**2-100**2 }{ 2 * 93.97 * 34.2 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+34.2**2-93.97**2 }{ 2 * 100 * 34.2 } ) = 70° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 70° = 20° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1606.97 }{ 114.09 } = 14.09 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 90° } = 50 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 93.97**2+2 * 34.2**2 - 100**2 } }{ 2 } = 50 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.2**2+2 * 100**2 - 93.97**2 } }{ 2 } = 58.115 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 93.97**2+2 * 100**2 - 34.2**2 } }{ 2 } = 95.513 ; ;
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