Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 12.04215945788   b = 6.40331242374   c = 5.83109518948

Area: T = 6.5
Perimeter: p = 24.27656707111
Semiperimeter: s = 12.13878353555

Angle ∠ A = α = 159.6243564786° = 159°37'25″ = 2.78659567693 rad
Angle ∠ B = β = 10.67697828045° = 10°40'11″ = 0.18662228404 rad
Angle ∠ C = γ = 9.70766524093° = 9°42'24″ = 0.16994130439 rad

Height: ha = 1.08795912381
Height: hb = 2.03302589046
Height: hc = 2.22994816069

Median: ma = 1.11880339887
Median: mb = 8.90222469074
Median: mc = 9.19223881554

Inradius: r = 0.53655155849
Circumradius: R = 17.29218731125

Vertex coordinates: A[0; -5] B[-5; -8] C[4; 0]
Centroid: CG[-0.33333333333; -4.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.84223519507; 0.53655155849]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 20.37664352138° = 20°22'35″ = 2.78659567693 rad
∠ B' = β' = 169.3330217196° = 169°19'49″ = 0.18662228404 rad
∠ C' = γ' = 170.2933347591° = 170°17'36″ = 0.16994130439 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{ (-5-4)**2 + (-8-0)**2 } ; ; a = sqrt{ 145 } = 12.04 ; ;

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{ (0-4)**2 + (-5-0)**2 } ; ; b = sqrt{ 41 } = 6.4 ; ;

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{ (0-(-5))**2 + (-5-(-8))**2 } ; ; c = sqrt{ 34 } = 5.83 ; ;


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 = 12.04 ; ; b = 6.4 ; ; c = 5.83 ; ;

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

p = a+b+c = 12.04+6.4+5.83 = 24.28 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.28 }{ 2 } = 12.14 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.14 * (12.14-12.04)(12.14-6.4)(12.14-5.83) } ; ; T = sqrt{ 42.25 } = 6.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.5 }{ 12.04 } = 1.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.5 }{ 6.4 } = 2.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.5 }{ 5.83 } = 2.23 ; ;

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{ 12.04**2-6.4**2-5.83**2 }{ 2 * 6.4 * 5.83 } ) = 159° 37'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.4**2-12.04**2-5.83**2 }{ 2 * 12.04 * 5.83 } ) = 10° 40'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.83**2-12.04**2-6.4**2 }{ 2 * 6.4 * 12.04 } ) = 9° 42'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.5 }{ 12.14 } = 0.54 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.04 }{ 2 * sin 159° 37'25" } = 17.29 ; ;




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