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 = 87.17879788708   b = 40   c = 100

Area: T = 1732.051080757
Perimeter: p = 227.1787978871
Semiperimeter: s = 113.5898989435

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 23.41332244464° = 23°24'48″ = 0.40986378551 rad
Angle ∠ C = γ = 96.58767755536° = 96°35'12″ = 1.68657572473 rad

Height: ha = 39.7365970712
Height: hb = 86.60325403784
Height: hc = 34.64110161514

Median: ma = 62.4549979984
Median: mb = 91.65215138991
Median: mc = 45.82657569496

Inradius: r = 15.2488404059
Circumradius: R = 50.33222295685

Vertex coordinates: A[100; 0] B[0; 0] C[80; 34.64110161514]
Centroid: CG[60; 11.54770053838]
Coordinates of the circumscribed circle: U[50; -5.77435026919]
Coordinates of the inscribed circle: I[73.58989894354; 15.2488404059]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 156.5876775554° = 156°35'12″ = 0.40986378551 rad
∠ C' = γ' = 83.41332244464° = 83°24'48″ = 1.68657572473 rad

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How did we calculate this triangle?

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

b = 40 ; ; c = 100 ; ; alpha = 60° ; ;

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{ 40**2+100**2 - 2 * 40 * 100 * cos(60° ) } ; ; a = 87.18 ; ;


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 = 87.18 ; ; b = 40 ; ; c = 100 ; ;

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

p = a+b+c = 87.18+40+100 = 227.18 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 227.18 }{ 2 } = 113.59 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 113.59 * (113.59-87.18)(113.59-40)(113.59-100) } ; ; T = sqrt{ 3000000 } = 1732.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1732.05 }{ 87.18 } = 39.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1732.05 }{ 40 } = 86.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1732.05 }{ 100 } = 34.64 ; ;

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{ 40**2+100**2-87.18**2 }{ 2 * 40 * 100 } ) = 60° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 87.18**2+100**2-40**2 }{ 2 * 87.18 * 100 } ) = 23° 24'48" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 87.18**2+40**2-100**2 }{ 2 * 87.18 * 40 } ) = 96° 35'12" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1732.05 }{ 113.59 } = 15.25 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 87.18 }{ 2 * sin 60° } = 50.33 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 100**2 - 87.18**2 } }{ 2 } = 62.45 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 87.18**2 - 40**2 } }{ 2 } = 91.652 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 87.18**2 - 100**2 } }{ 2 } = 45.826 ; ;
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