Triangle calculator

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

You have entered side a, b and angle γ.

Acute scalene triangle.

Sides: a = 91   b = 68   c = 112.6465501543

Area: T = 3093.529876881
Perimeter: p = 271.6465501543
Semiperimeter: s = 135.8232750772

Angle ∠ A = α = 53.87438635249° = 53°52'26″ = 0.94402762993 rad
Angle ∠ B = β = 37.12661364751° = 37°7'34″ = 0.648797332 rad
Angle ∠ C = γ = 89° = 1.55333430343 rad

Height: ha = 67.99896432706
Height: hb = 90.98661402592
Height: hc = 54.92550298757

Median: ma = 81.15657423043
Median: mb = 96.5876771915
Median: mc = 57.27334471244

Inradius: r = 22.77662193833
Circumradius: R = 56.33113303061

Vertex coordinates: A[112.6465501543; 0] B[0; 0] C[72.55550900568; 54.92550298757]
Centroid: CG[61.73435305334; 18.30883432919]
Coordinates of the circumscribed circle: U[56.32327507716; 0.98331172717]
Coordinates of the inscribed circle: I[67.82327507716; 22.77662193833]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.1266136475° = 126°7'34″ = 0.94402762993 rad
∠ B' = β' = 142.8743863525° = 142°52'26″ = 0.648797332 rad
∠ C' = γ' = 91° = 1.55333430343 rad

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How did we calculate this triangle?

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

a = 91 ; ; b = 68 ; ; gamma = 89° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 91**2+68**2 - 2 * 91 * 68 * cos 89° } ; ; c = 112.65 ; ;
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 = 91 ; ; b = 68 ; ; c = 112.65 ; ;

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

p = a+b+c = 91+68+112.65 = 271.65 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 271.65 }{ 2 } = 135.82 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.82 * (135.82-91)(135.82-68)(135.82-112.65) } ; ; T = sqrt{ 9569920.24 } = 3093.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3093.53 }{ 91 } = 67.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3093.53 }{ 68 } = 90.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3093.53 }{ 112.65 } = 54.93 ; ;

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{ 68**2+112.65**2-91**2 }{ 2 * 68 * 112.65 } ) = 53° 52'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 91**2+112.65**2-68**2 }{ 2 * 91 * 112.65 } ) = 37° 7'34" ; ;
 gamma = 180° - alpha - beta = 180° - 53° 52'26" - 37° 7'34" = 89° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3093.53 }{ 135.82 } = 22.78 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 91 }{ 2 * sin 53° 52'26" } = 56.33 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 68**2+2 * 112.65**2 - 91**2 } }{ 2 } = 81.156 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 112.65**2+2 * 91**2 - 68**2 } }{ 2 } = 96.587 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 68**2+2 * 91**2 - 112.65**2 } }{ 2 } = 57.273 ; ;
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