Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 3.60655512755   b = 4.4722135955   c = 6.08327625303

Area: T = 8
Perimeter: p = 14.16604497608
Semiperimeter: s = 7.08802248804

Angle ∠ A = α = 36.02773733851° = 36°1'39″ = 0.62987962864 rad
Angle ∠ B = β = 46.8487610266° = 46°50'51″ = 0.81876450458 rad
Angle ∠ C = γ = 97.12550163489° = 97°7'30″ = 1.69551513213 rad

Height: ha = 4.43876015698
Height: hb = 3.5787708764
Height: hc = 2.63303837969

Median: ma = 5.02549378106
Median: mb = 4.4722135955
Median: mc = 2.69325824036

Inradius: r = 1.13299076138
Circumradius: R = 3.06550499588

Vertex coordinates: A[1; 3] B[2; -3] C[-1; -1]
Centroid: CG[0.66766666667; -0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.0599288388; 1.13299076138]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.9732626615° = 143°58'21″ = 0.62987962864 rad
∠ B' = β' = 133.1522389734° = 133°9'9″ = 0.81876450458 rad
∠ C' = γ' = 82.87549836511° = 82°52'30″ = 1.69551513213 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{ (2-(-1))**2 + (-3-(-1))**2 } ; ; a = sqrt{ 13 } = 3.61 ; ;

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-(-1))**2 } ; ; b = sqrt{ 20 } = 4.47 ; ;

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{ (1-2)**2 + (3-(-3))**2 } ; ; c = sqrt{ 37 } = 6.08 ; ;


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 = 3.61 ; ; b = 4.47 ; ; c = 6.08 ; ;

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

p = a+b+c = 3.61+4.47+6.08 = 14.16 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.16 }{ 2 } = 7.08 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.08 * (7.08-3.61)(7.08-4.47)(7.08-6.08) } ; ; 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 }{ 3.61 } = 4.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8 }{ 4.47 } = 3.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8 }{ 6.08 } = 2.63 ; ;

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{ 3.61**2-4.47**2-6.08**2 }{ 2 * 4.47 * 6.08 } ) = 36° 1'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.47**2-3.61**2-6.08**2 }{ 2 * 3.61 * 6.08 } ) = 46° 50'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.08**2-3.61**2-4.47**2 }{ 2 * 4.47 * 3.61 } ) = 97° 7'30" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8 }{ 7.08 } = 1.13 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.61 }{ 2 * sin 36° 1'39" } = 3.07 ; ;




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