Triangle calculator

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

You have entered side b, c and angle α.

Obtuse scalene triangle.

Sides: a = 68.40770692634   b = 53.2   c = 100

Area: T = 1709.815504177
Perimeter: p = 221.6077069263
Semiperimeter: s = 110.8043534632

Angle ∠ A = α = 40° = 0.69881317008 rad
Angle ∠ B = β = 29.99330041258° = 29°59'35″ = 0.52334766746 rad
Angle ∠ C = γ = 110.0076995874° = 110°25″ = 1.92199842782 rad

Height: ha = 49.98994253672
Height: hb = 64.27987609687
Height: hc = 34.19663008353

Median: ma = 72.42440168639
Median: mb = 81.43883420915
Median: mc = 35.42443357398

Inradius: r = 15.43110514322
Circumradius: R = 53.21112537894

Vertex coordinates: A[100; 0] B[0; 0] C[59.24664356261; 34.19663008353]
Centroid: CG[53.08221452087; 11.39987669451]
Coordinates of the circumscribed circle: U[50; -18.20554258352]
Coordinates of the inscribed circle: I[57.60435346317; 15.43110514322]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140° = 0.69881317008 rad
∠ B' = β' = 150.0076995874° = 150°25″ = 0.52334766746 rad
∠ C' = γ' = 69.99330041258° = 69°59'35″ = 1.92199842782 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 53.2 ; ; c = 100 ; ; alpha = 40° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 53.2**2+100**2 - 2 * 53.2 * 100 * cos 40° } ; ; a = 68.41 ; ;


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 = 68.41 ; ; b = 53.2 ; ; c = 100 ; ;

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

p = a+b+c = 68.41+53.2+100 = 221.61 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 221.61 }{ 2 } = 110.8 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.8 * (110.8-68.41)(110.8-53.2)(110.8-100) } ; ; T = sqrt{ 2923467.48 } = 1709.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1709.82 }{ 68.41 } = 49.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1709.82 }{ 53.2 } = 64.28 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1709.82 }{ 100 } = 34.2 ; ;

7. 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{ 53.2**2+100**2-68.41**2 }{ 2 * 53.2 * 100 } ) = 40° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 68.41**2+100**2-53.2**2 }{ 2 * 68.41 * 100 } ) = 29° 59'35" ; ; gamma = 180° - alpha - beta = 180° - 40° - 29° 59'35" = 110° 25" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1709.82 }{ 110.8 } = 15.43 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 68.41 }{ 2 * sin 40° } = 53.21 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 53.2**2+2 * 100**2 - 68.41**2 } }{ 2 } = 72.424 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 68.41**2 - 53.2**2 } }{ 2 } = 81.438 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 53.2**2+2 * 68.41**2 - 100**2 } }{ 2 } = 35.424 ; ;
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