Triangle calculator

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

You have entered side a, c and angle β.

Obtuse scalene triangle.

Sides: a = 1.2   b = 0.91105925113   c = 2.1

Area: T = 0.11098162359
Perimeter: p = 4.21105925113
Semiperimeter: s = 2.10552962556

Angle ∠ A = α = 6.5955311683° = 6°35'43″ = 0.11551099041 rad
Angle ∠ B = β = 5° = 0.08772664626 rad
Angle ∠ C = γ = 168.4054688317° = 168°24'17″ = 2.93992162869 rad

Height: ha = 0.18330270598
Height: hb = 0.24111973182
Height: hc = 0.10545868913

Median: ma = 1.50331930551
Median: mb = 1.64985464263
Median: mc = 0.17991350351

Inradius: r = 0.0522161892
Circumradius: R = 5.22439386791

Vertex coordinates: A[2.1; 0] B[0; 0] C[1.19554336377; 0.10545868913]
Centroid: CG[1.09884778792; 0.03548622971]
Coordinates of the circumscribed circle: U[1.05; -5.11773269705]
Coordinates of the inscribed circle: I[1.19547037444; 0.0522161892]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173.4054688317° = 173°24'17″ = 0.11551099041 rad
∠ B' = β' = 175° = 0.08772664626 rad
∠ C' = γ' = 11.5955311683° = 11°35'43″ = 2.93992162869 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and angle β.

a = 1.2 ; ; c = 2.1 ; ; beta = 5° ; ;

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

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 1.2**2+2.1**2 - 2 * 1.2 * 2.1 * cos 5° } ; ; b = 0.91 ; ;


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 = 1.2 ; ; b = 0.91 ; ; c = 2.1 ; ;

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

p = a+b+c = 1.2+0.91+2.1 = 4.21 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.21 }{ 2 } = 2.11 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.11 * (2.11-1.2)(2.11-0.91)(2.11-2.1) } ; ; T = sqrt{ 0.01 } = 0.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.11 }{ 1.2 } = 0.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.11 }{ 0.91 } = 0.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.11 }{ 2.1 } = 0.1 ; ;

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{ 0.91**2+2.1**2-1.2**2 }{ 2 * 0.91 * 2.1 } ) = 6° 35'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.2**2+2.1**2-0.91**2 }{ 2 * 1.2 * 2.1 } ) = 5° ; ; gamma = 180° - alpha - beta = 180° - 6° 35'43" - 5° = 168° 24'17" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.11 }{ 2.11 } = 0.05 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1.2 }{ 2 * sin 6° 35'43" } = 5.22 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.91**2+2 * 2.1**2 - 1.2**2 } }{ 2 } = 1.503 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.1**2+2 * 1.2**2 - 0.91**2 } }{ 2 } = 1.649 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.91**2+2 * 1.2**2 - 2.1**2 } }{ 2 } = 0.179 ; ;
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