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 = 76   b = 43.81880567745   c = 44

Area: T = 836
Perimeter: p = 163.8188056775
Semiperimeter: s = 81.90990283872

Angle ∠ A = α = 119.8632549584° = 119°51'45″ = 2.09219961401 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 30.13774504164° = 30°8'15″ = 0.52659977379 rad

Height: ha = 22
Height: hb = 38.15877852392
Height: hc = 38

Median: ma = 222.0002511291
Median: mb = 58.10333086418
Median: mc = 588.0000952563

Inradius: r = 10.20664450826
Circumradius: R = 43.81880567745

Vertex coordinates: A[44; 0] B[0; 0] C[65.81879306876; 38]
Centroid: CG[36.60659768959; 12.66766666667]
Coordinates of the circumscribed circle: U[22; 37.89548822335]
Coordinates of the inscribed circle: I[38.09109716128; 10.20664450826]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 60.13774504164° = 60°8'15″ = 2.09219961401 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 149.8632549584° = 149°51'45″ = 0.52659977379 rad

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How did we calculate this triangle?

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

a = 76 ; ; c = 44 ; ; beta = 30° ; ;

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{ 76**2+44**2 - 2 * 76 * 44 * cos 30° } ; ; b = 43.82 ; ;


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 = 76 ; ; b = 43.82 ; ; c = 44 ; ;

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

p = a+b+c = 76+43.82+44 = 163.82 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 163.82 }{ 2 } = 81.91 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.91 * (81.91-76)(81.91-43.82)(81.91-44) } ; ; T = sqrt{ 698896 } = 836 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 836 }{ 76 } = 22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 836 }{ 43.82 } = 38.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 836 }{ 44 } = 38 ; ;

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{ 43.82**2+44**2-76**2 }{ 2 * 43.82 * 44 } ) = 119° 51'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 76**2+44**2-43.82**2 }{ 2 * 76 * 44 } ) = 30° ; ; gamma = 180° - alpha - beta = 180° - 119° 51'45" - 30° = 30° 8'15" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 836 }{ 81.91 } = 10.21 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 76 }{ 2 * sin 119° 51'45" } = 43.82 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.82**2+2 * 44**2 - 76**2 } }{ 2 } = 22 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 44**2+2 * 76**2 - 43.82**2 } }{ 2 } = 58.103 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.82**2+2 * 76**2 - 44**2 } }{ 2 } = 58 ; ;
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