Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 7.61657731059   b = 3.60655512755   c = 6.40331242374

Area: T = 11.5
Perimeter: p = 17.62444486188
Semiperimeter: s = 8.81222243094

Angle ∠ A = α = 94.97697407281° = 94°58'11″ = 1.65875346655 rad
Angle ∠ B = β = 28.14216012323° = 28°8'30″ = 0.49111635983 rad
Angle ∠ C = γ = 56.88986580396° = 56°53'19″ = 0.99328943898 rad

Height: ha = 3.02200479558
Height: hb = 6.37990522566
Height: hc = 3.59219965234

Median: ma = 3.53655339059
Median: mb = 6.80107352544
Median: mc = 5.02549378106

Inradius: r = 1.30550053648
Circumradius: R = 3.82222559872

Vertex coordinates: A[1; 3] B[-4; -1] C[-1; 6]
Centroid: CG[-1.33333333333; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.44397926386; 1.30550053648]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 85.03302592719° = 85°1'49″ = 1.65875346655 rad
∠ B' = β' = 151.8588398768° = 151°51'30″ = 0.49111635983 rad
∠ C' = γ' = 123.111134196° = 123°6'41″ = 0.99328943898 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{ (-4-(-1))**2 + (-1-6)**2 } ; ; a = sqrt{ 58 } = 7.62 ; ;

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{ (1-(-1))**2 + (3-6)**2 } ; ; b = sqrt{ 13 } = 3.61 ; ;

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{ (1-(-4))**2 + (3-(-1))**2 } ; ; c = sqrt{ 41 } = 6.4 ; ;


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 = 7.62 ; ; b = 3.61 ; ; c = 6.4 ; ;

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

p = a+b+c = 7.62+3.61+6.4 = 17.62 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.62 }{ 2 } = 8.81 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.81 * (8.81-7.62)(8.81-3.61)(8.81-6.4) } ; ; T = sqrt{ 132.25 } = 11.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.5 }{ 7.62 } = 3.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.5 }{ 3.61 } = 6.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.5 }{ 6.4 } = 3.59 ; ;

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{ 7.62**2-3.61**2-6.4**2 }{ 2 * 3.61 * 6.4 } ) = 94° 58'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.61**2-7.62**2-6.4**2 }{ 2 * 7.62 * 6.4 } ) = 28° 8'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.4**2-7.62**2-3.61**2 }{ 2 * 3.61 * 7.62 } ) = 56° 53'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.5 }{ 8.81 } = 1.31 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.62 }{ 2 * sin 94° 58'11" } = 3.82 ; ;




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