Triangle calculator

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

You have entered side a, b and angle γ.

Obtuse isosceles triangle.

Sides: a = 69.5   b = 69.5   c = 104.105484519

Area: T = 2397.123302448
Perimeter: p = 243.105484519
Semiperimeter: s = 121.5522422595

Angle ∠ A = α = 41.5° = 41°30' = 0.72443116396 rad
Angle ∠ B = β = 41.5° = 41°30' = 0.72443116396 rad
Angle ∠ C = γ = 97° = 1.69329693744 rad

Height: ha = 68.98219575391
Height: hb = 68.98219575391
Height: hc = 46.0522093351

Median: ma = 81.40331442635
Median: mb = 81.40331442635
Median: mc = 46.0522093351

Inradius: r = 19.72108987967
Circumradius: R = 52.44333272032

Vertex coordinates: A[104.105484519; 0] B[0; 0] C[52.05224225948; 46.0522093351]
Centroid: CG[52.05224225948; 15.35106977837]
Coordinates of the circumscribed circle: U[52.05224225948; -6.39112338522]
Coordinates of the inscribed circle: I[52.05224225948; 19.72108987967]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.5° = 138°30' = 0.72443116396 rad
∠ B' = β' = 138.5° = 138°30' = 0.72443116396 rad
∠ C' = γ' = 83° = 1.69329693744 rad

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How did we calculate this triangle?

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

a = 69.5 ; ; b = 69.5 ; ; gamma = 97° ; ;

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{ 69.5**2+69.5**2 - 2 * 69.5 * 69.5 * cos 97° } ; ; c = 104.1 ; ;


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 = 69.5 ; ; b = 69.5 ; ; c = 104.1 ; ;

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

p = a+b+c = 69.5+69.5+104.1 = 243.1 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 243.1 }{ 2 } = 121.55 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 121.55 * (121.55-69.5)(121.55-69.5)(121.55-104.1) } ; ; T = sqrt{ 5746198.79 } = 2397.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2397.12 }{ 69.5 } = 68.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2397.12 }{ 69.5 } = 68.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2397.12 }{ 104.1 } = 46.05 ; ;

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{ 69.5**2+104.1**2-69.5**2 }{ 2 * 69.5 * 104.1 } ) = 41° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 69.5**2+104.1**2-69.5**2 }{ 2 * 69.5 * 104.1 } ) = 41° 30' ; ; gamma = 180° - alpha - beta = 180° - 41° 30' - 41° 30' = 97° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2397.12 }{ 121.55 } = 19.72 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 69.5 }{ 2 * sin 41° 30' } = 52.44 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 69.5**2+2 * 104.1**2 - 69.5**2 } }{ 2 } = 81.403 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 104.1**2+2 * 69.5**2 - 69.5**2 } }{ 2 } = 81.403 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 69.5**2+2 * 69.5**2 - 104.1**2 } }{ 2 } = 46.052 ; ;
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