Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 6.32545553203   b = 8.6022325267   c = 3.16222776602

Area: T = 8
Perimeter: p = 18.08991582475
Semiperimeter: s = 9.04545791238

Angle ∠ A = α = 36.02773733851° = 36°1'39″ = 0.62987962864 rad
Angle ∠ B = β = 126.8769897646° = 126°52'12″ = 2.21442974356 rad
Angle ∠ C = γ = 17.10327289691° = 17°6'10″ = 0.29884989316 rad

Height: ha = 2.53298221281
Height: hb = 1.86599622199
Height: hc = 5.06596442563

Median: ma = 5.65768542495
Median: mb = 2.55495097568
Median: mc = 7.38224115301

Inradius: r = 0.88545077135
Circumradius: R = 5.37664532919

Vertex coordinates: A[-4; -2] B[-3; -5] C[3; -7]
Centroid: CG[-1.33333333333; -4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.66333807851; 0.88545077135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.9732626615° = 143°58'21″ = 0.62987962864 rad
∠ B' = β' = 53.13301023542° = 53°7'48″ = 2.21442974356 rad
∠ C' = γ' = 162.8977271031° = 162°53'50″ = 0.29884989316 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{ (-3-3)**2 + (-5-(-7))**2 } ; ; a = sqrt{ 40 } = 6.32 ; ;

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{ (-4-3)**2 + (-2-(-7))**2 } ; ; b = sqrt{ 74 } = 8.6 ; ;

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{ (-4-(-3))**2 + (-2-(-5))**2 } ; ; c = sqrt{ 10 } = 3.16 ; ;


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 = 6.32 ; ; b = 8.6 ; ; c = 3.16 ; ;

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

p = a+b+c = 6.32+8.6+3.16 = 18.09 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.09 }{ 2 } = 9.04 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.04 * (9.04-6.32)(9.04-8.6)(9.04-3.16) } ; ; T = sqrt{ 64 } = 8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8 }{ 6.32 } = 2.53 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8 }{ 8.6 } = 1.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8 }{ 3.16 } = 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{ 6.32**2-8.6**2-3.16**2 }{ 2 * 8.6 * 3.16 } ) = 36° 1'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.6**2-6.32**2-3.16**2 }{ 2 * 6.32 * 3.16 } ) = 126° 52'12" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.16**2-6.32**2-8.6**2 }{ 2 * 8.6 * 6.32 } ) = 17° 6'10" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8 }{ 9.04 } = 0.88 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.32 }{ 2 * sin 36° 1'39" } = 5.38 ; ;




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