Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 10.77703296143   b = 5.38551648071   c = 12.04215945788

Area: T = 29
Perimeter: p = 28.19770890002
Semiperimeter: s = 14.09985445001

Angle ∠ A = α = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ B = β = 26.56550511771° = 26°33'54″ = 0.4643647609 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 5.38551648071
Height: hb = 10.77703296143
Height: hc = 4.81766378315

Median: ma = 7.61657731059
Median: mb = 11.10218016556
Median: mc = 6.02107972894

Inradius: r = 2.05769499213
Circumradius: R = 6.02107972894

Vertex coordinates: A[-6; 3] B[3; -5] C[-1; 5]
Centroid: CG[-1.33333333333; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[4.11438998426; 2.05769499213]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 153.4354948823° = 153°26'6″ = 0.4643647609 rad
∠ C' = γ' = 90° = 1.57107963268 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-(-1))**2 + (-5-5)**2 } ; ; a = sqrt{ 116 } = 10.77 ; ;

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{ (-6-(-1))**2 + (3-5)**2 } ; ; b = sqrt{ 29 } = 5.39 ; ;

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{ (-6-3)**2 + (3-(-5))**2 } ; ; c = sqrt{ 145 } = 12.04 ; ;


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 = 10.77 ; ; b = 5.39 ; ; c = 12.04 ; ;

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

p = a+b+c = 10.77+5.39+12.04 = 28.2 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28.2 }{ 2 } = 14.1 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.1 * (14.1-10.77)(14.1-5.39)(14.1-12.04) } ; ; T = sqrt{ 841 } = 29 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29 }{ 10.77 } = 5.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29 }{ 5.39 } = 10.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29 }{ 12.04 } = 4.82 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.39**2+12.04**2-10.77**2 }{ 2 * 5.39 * 12.04 } ) = 63° 26'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.77**2+12.04**2-5.39**2 }{ 2 * 10.77 * 12.04 } ) = 26° 33'54" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 10.77**2+5.39**2-12.04**2 }{ 2 * 10.77 * 5.39 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29 }{ 14.1 } = 2.06 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.77 }{ 2 * sin 63° 26'6" } = 6.02 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 12.04**2 - 10.77**2 } }{ 2 } = 7.616 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.04**2+2 * 10.77**2 - 5.39**2 } }{ 2 } = 11.102 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 10.77**2 - 12.04**2 } }{ 2 } = 6.021 ; ;
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