Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 5.83109518948   b = 6.08327625303   c = 2.23660679775

Area: T = 6.5
Perimeter: p = 14.15497824026
Semiperimeter: s = 7.07548912013

Angle ∠ A = α = 72.89772710309° = 72°53'50″ = 1.27222973952 rad
Angle ∠ B = β = 85.6011294645° = 85°36'5″ = 1.49440244355 rad
Angle ∠ C = γ = 21.5011434324° = 21°30'5″ = 0.37552708229 rad

Height: ha = 2.22994816069
Height: hb = 2.1377186835
Height: hc = 5.81437767415

Median: ma = 3.53655339059
Median: mb = 3.20215621187
Median: mc = 5.85223499554

Inradius: r = 0.91987420435
Circumradius: R = 3.05503661629

Vertex coordinates: A[3; 1] B[2; -1] C[-3; 2]
Centroid: CG[0.66766666667; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.07106724649; 0.91987420435]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.1032728969° = 107°6'10″ = 1.27222973952 rad
∠ B' = β' = 94.3998705355° = 94°23'55″ = 1.49440244355 rad
∠ C' = γ' = 158.4998565676° = 158°29'55″ = 0.37552708229 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-(-3))**2 + (-1-2)**2 } ; ; a = sqrt{ 34 } = 5.83 ; ;

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

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{ (3-2)**2 + (1-(-1))**2 } ; ; c = sqrt{ 5 } = 2.24 ; ;


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 = 5.83 ; ; b = 6.08 ; ; c = 2.24 ; ;

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

p = a+b+c = 5.83+6.08+2.24 = 14.15 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.15 }{ 2 } = 7.07 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.07 * (7.07-5.83)(7.07-6.08)(7.07-2.24) } ; ; 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 }{ 5.83 } = 2.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.5 }{ 6.08 } = 2.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.5 }{ 2.24 } = 5.81 ; ;

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{ 5.83**2-6.08**2-2.24**2 }{ 2 * 6.08 * 2.24 } ) = 72° 53'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.08**2-5.83**2-2.24**2 }{ 2 * 5.83 * 2.24 } ) = 85° 36'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.24**2-5.83**2-6.08**2 }{ 2 * 6.08 * 5.83 } ) = 21° 30'5" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.5 }{ 7.07 } = 0.92 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.83 }{ 2 * sin 72° 53'50" } = 3.05 ; ;




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