Triangle calculator

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

You have entered side b, angle α and angle β.

Acute scalene triangle.

Sides: a = 80.18766663967   b = 68   c = 96.68546582675

Area: T = 2692.781080602
Perimeter: p = 244.8711324664
Semiperimeter: s = 122.4365662332

Angle ∠ A = α = 55° = 0.96599310886 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 81° = 1.41437166941 rad

Height: ha = 67.16328071605
Height: hb = 79.19994354712
Height: hc = 55.70223390117

Median: ma = 73.33881633612
Median: mb = 82.05443253342
Median: mc = 56.4879818943

Inradius: r = 21.99334352029
Circumradius: R = 48.94549223473

Vertex coordinates: A[96.68546582675; 0] B[0; 0] C[57.68114605956; 55.70223390117]
Centroid: CG[51.45553729544; 18.56774463372]
Coordinates of the circumscribed circle: U[48.34223291337; 7.65766727438]
Coordinates of the inscribed circle: I[54.43656623321; 21.99334352029]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125° = 0.96599310886 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 99° = 1.41437166941 rad

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How did we calculate this triangle?

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

b = 68 ; ; alpha = 55° ; ; beta = 44° ; ;

2. From angle α and angle β we calculate angle γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 55 ° - 44 ° = 81 ° ; ;

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

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 68 * fraction{ sin 55° }{ sin 44° } = 80.19 ; ;

4. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 68**2+80.19**2 - 2 * 68 * 80.19 * cos 81° } ; ; c = 96.68 ; ;


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 = 80.19 ; ; b = 68 ; ; c = 96.68 ; ;

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

p = a+b+c = 80.19+68+96.68 = 244.87 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 244.87 }{ 2 } = 122.44 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 122.44 * (122.44-80.19)(122.44-68)(122.44-96.68) } ; ; T = sqrt{ 7251068.47 } = 2692.78 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2692.78 }{ 80.19 } = 67.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2692.78 }{ 68 } = 79.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2692.78 }{ 96.68 } = 55.7 ; ;

9. 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{ 68**2+96.68**2-80.19**2 }{ 2 * 68 * 96.68 } ) = 55° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 80.19**2+96.68**2-68**2 }{ 2 * 80.19 * 96.68 } ) = 44° ; ; gamma = 180° - alpha - beta = 180° - 55° - 44° = 81° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2692.78 }{ 122.44 } = 21.99 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 80.19 }{ 2 * sin 55° } = 48.94 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 68**2+2 * 96.68**2 - 80.19**2 } }{ 2 } = 73.338 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 96.68**2+2 * 80.19**2 - 68**2 } }{ 2 } = 82.054 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 68**2+2 * 80.19**2 - 96.68**2 } }{ 2 } = 56.48 ; ;
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