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 scalene triangle.

Sides: a = 12.77   b = 6.72   c = 6.44331288095

Area: T = 10.1911161856
Perimeter: p = 25.93331288095
Semiperimeter: s = 12.96765644048

Angle ∠ A = α = 151.9177199843° = 151°55'2″ = 2.65114553277 rad
Angle ∠ B = β = 14.34328001573° = 14°20'34″ = 0.25503290867 rad
Angle ∠ C = γ = 13.74° = 13°44'24″ = 0.24398082392 rad

Height: ha = 1.59661099226
Height: hb = 3.03330838857
Height: hc = 3.16334201821

Median: ma = 1.60224760304
Median: mb = 9.54395914183
Median: mc = 9.68217959484

Inradius: r = 0.78659569843
Circumradius: R = 13.56435475308

Vertex coordinates: A[6.44331288095; 0] B[0; 0] C[12.37219712557; 3.16334201821]
Centroid: CG[6.27217000217; 1.0544473394]
Coordinates of the circumscribed circle: U[3.22215644048; 13.17554068023]
Coordinates of the inscribed circle: I[6.24765644048; 0.78659569843]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 28.08328001573° = 28°4'58″ = 2.65114553277 rad
∠ B' = β' = 165.6577199843° = 165°39'26″ = 0.25503290867 rad
∠ C' = γ' = 166.26° = 166°15'36″ = 0.24398082392 rad

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How did we calculate this triangle?

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

a = 12.77 ; ; b = 6.72 ; ; gamma = 13.74° ; ;

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{ 12.77**2+6.72**2 - 2 * 12.77 * 6.72 * cos(13° 44'24") } ; ; c = 6.44 ; ;


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 = 12.77 ; ; b = 6.72 ; ; c = 6.44 ; ;

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

p = a+b+c = 12.77+6.72+6.44 = 25.93 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.93 }{ 2 } = 12.97 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.97 * (12.97-12.77)(12.97-6.72)(12.97-6.44) } ; ; T = sqrt{ 103.86 } = 10.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.19 }{ 12.77 } = 1.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.19 }{ 6.72 } = 3.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.19 }{ 6.44 } = 3.16 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12.77**2-6.72**2-6.44**2 }{ 2 * 6.72 * 6.44 } ) = 151° 55'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.72**2-12.77**2-6.44**2 }{ 2 * 12.77 * 6.44 } ) = 14° 20'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.44**2-12.77**2-6.72**2 }{ 2 * 6.72 * 12.77 } ) = 13° 44'24" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.19 }{ 12.97 } = 0.79 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.77 }{ 2 * sin 151° 55'2" } = 13.56 ; ;




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