Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.48552813742   b = 21.09550231097   c = 27.29546881279

Area: T = 69
Perimeter: p = 56.87549926119
Semiperimeter: s = 28.43774963059

Angle ∠ A = α = 13.86773010525° = 13°52'2″ = 0.24220300617 rad
Angle ∠ B = β = 36.57330309785° = 36°34'23″ = 0.6388319808 rad
Angle ∠ C = γ = 129.5659667969° = 129°33'35″ = 2.26112427838 rad

Height: ha = 16.26334559673
Height: hb = 6.54218273913
Height: hc = 5.05659288076

Median: ma = 24.02108242989
Median: mb = 17.24109396496
Median: mc = 8.5

Inradius: r = 2.42663739416
Circumradius: R = 17.70217135223

Vertex coordinates: A[-19; -8] B[8; -12] C[2; -6]
Centroid: CG[-3; -8.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.27703300951; 2.42663739416]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.1332698948° = 166°7'58″ = 0.24220300617 rad
∠ B' = β' = 143.4276969021° = 143°25'37″ = 0.6388319808 rad
∠ C' = γ' = 50.4440332031° = 50°26'25″ = 2.26112427838 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (8-2)**2 + (-12-(-6))**2 } ; ; a = sqrt{ 72 } = 8.49 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (-19-2)**2 + (-8-(-6))**2 } ; ; b = sqrt{ 445 } = 21.1 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (-19-8)**2 + (-8-(-12))**2 } ; ; c = sqrt{ 745 } = 27.29 ; ;


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 = 8.49 ; ; b = 21.1 ; ; c = 27.29 ; ;

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

p = a+b+c = 8.49+21.1+27.29 = 56.87 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56.87 }{ 2 } = 28.44 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.44 * (28.44-8.49)(28.44-21.1)(28.44-27.29) } ; ; T = sqrt{ 4761 } = 69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69 }{ 8.49 } = 16.26 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69 }{ 21.1 } = 6.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69 }{ 27.29 } = 5.06 ; ;

8. 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{ 8.49**2-21.1**2-27.29**2 }{ 2 * 21.1 * 27.29 } ) = 13° 52'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21.1**2-8.49**2-27.29**2 }{ 2 * 8.49 * 27.29 } ) = 36° 34'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27.29**2-8.49**2-21.1**2 }{ 2 * 21.1 * 8.49 } ) = 129° 33'35" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69 }{ 28.44 } = 2.43 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.49 }{ 2 * sin 13° 52'2" } = 17.7 ; ;




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