Triangle calculator

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

You have entered side a, b and angle β.

Acute scalene triangle.

Sides: a = 15.4   b = 25.8   c = 27.25875591413

Area: T = 194.6600319381
Perimeter: p = 68.45875591413
Semiperimeter: s = 34.22987795707

Angle ∠ A = α = 33.60330112752° = 33°36'11″ = 0.58664831853 rad
Angle ∠ B = β = 68° = 1.18768238914 rad
Angle ∠ C = γ = 78.39769887248° = 78°23'49″ = 1.36882855769 rad

Height: ha = 25.27327687507
Height: hb = 15.0855296076
Height: hc = 14.27986313603

Median: ma = 25.39771901039
Median: mb = 17.9990477069
Median: mc = 16.29989682929

Inradius: r = 5.6855283607
Circumradius: R = 13.91330981805

Vertex coordinates: A[27.25875591413; 0] B[0; 0] C[5.76989415386; 14.27986313603]
Centroid: CG[11.009883356; 4.76595437868]
Coordinates of the circumscribed circle: U[13.62987795707; 2.79883331459]
Coordinates of the inscribed circle: I[8.42987795707; 5.6855283607]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.3976988725° = 146°23'49″ = 0.58664831853 rad
∠ B' = β' = 112° = 1.18768238914 rad
∠ C' = γ' = 101.6033011275° = 101°36'11″ = 1.36882855769 rad

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How did we calculate this triangle?

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

a = 15.4 ; ; b = 25.8 ; ; beta = 68° ; ;

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 25.8**2 = 15.4**2 + c**2 - 2 * 15.4 * c * cos 68° ; ; ; ; ; ; c**2 -11.538c -428.48 =0 ; ; p=1; q=-11.538; r=-428.48 ; ; D = q**2 - 4pr = 11.538**2 - 4 * 1 * (-428.48) = 1847.0427459 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 11.54 ± sqrt{ 1847.04 } }{ 2 } ; ; c_{1,2} = 5.76894154 ± 21.4886176027 ; ; c_{1} = 27.2575591427 ; ; c_{2} = -15.7196760627 ; ;
 ; ; text{ Factored form: } ; ; (c -27.2575591427) (c +15.7196760627) = 0 ; ; ; ; c > 0 ; ; ; ; c = 27.258 ; ;
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 = 15.4 ; ; b = 25.8 ; ; c = 27.26 ; ;

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

p = a+b+c = 15.4+25.8+27.26 = 68.46 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68.46 }{ 2 } = 34.23 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.23 * (34.23-15.4)(34.23-25.8)(34.23-27.26) } ; ; T = sqrt{ 37869.28 } = 194.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 194.6 }{ 15.4 } = 25.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 194.6 }{ 25.8 } = 15.09 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 194.6 }{ 27.26 } = 14.28 ; ;

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{ 25.8**2+27.26**2-15.4**2 }{ 2 * 25.8 * 27.26 } ) = 33° 36'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.4**2+27.26**2-25.8**2 }{ 2 * 15.4 * 27.26 } ) = 68° ; ; gamma = 180° - alpha - beta = 180° - 33° 36'11" - 68° = 78° 23'49" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 194.6 }{ 34.23 } = 5.69 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.4 }{ 2 * sin 33° 36'11" } = 13.91 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.8**2+2 * 27.26**2 - 15.4**2 } }{ 2 } = 25.397 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.26**2+2 * 15.4**2 - 25.8**2 } }{ 2 } = 17.99 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25.8**2+2 * 15.4**2 - 27.26**2 } }{ 2 } = 16.299 ; ;
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