Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 7.07110678119   b = 8.12440384046   c = 3.74216573868

Area: T = 13.21993040664
Perimeter: p = 18.93767636033
Semiperimeter: s = 9.46883818016

Angle ∠ A = α = 60.43114994029° = 60°25'53″ = 1.05547286365 rad
Angle ∠ B = β = 92.16660928562° = 92°9'58″ = 1.6098601779 rad
Angle ∠ C = γ = 27.40224077409° = 27°24'9″ = 0.47882622381 rad

Height: ha = 3.73989838192
Height: hb = 3.25443676945
Height: hc = 7.0666015244

Median: ma = 5.24440442409
Median: mb = 3.9377003937
Median: mc = 7.38224115301

Inradius: r = 1.39661524095
Circumradius: R = 4.0654923757

Vertex coordinates: A[2; -3; 4] B[-1; -2; 2] C[3; 1; -3]
Centroid: CG[1.33333333333; -1.33333333333; 1]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119.5698500597° = 119°34'7″ = 1.05547286365 rad
∠ B' = β' = 87.83439071438° = 87°50'2″ = 1.6098601779 rad
∠ C' = γ' = 152.5987592259° = 152°35'51″ = 0.47882622381 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 + ( beta _z- gamma _z)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 + ( beta _z- gamma _z)**2 } ; ; a = sqrt{ (-1-3)**2 + (-2-1)**2 + (2 - (-3))**2 } ; ; a = sqrt{ 50 } = 7.07 ; ;

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 + ( alpha _z- gamma _z)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 + ( alpha _z- gamma _z)**2 } ; ; b = sqrt{ (2-3)**2 + (-3-1)**2 + (4 - (-3))**2 } ; ; b = sqrt{ 66 } = 8.12 ; ;

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 + ( alpha _z- beta _z)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 + ( alpha _z- beta _z)**2 } ; ; c = sqrt{ (2-(-1))**2 + (-3-(-2))**2 + (4 - 2)**2 } ; ; c = sqrt{ 14 } = 3.74 ; ;


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 = 7.07 ; ; b = 8.12 ; ; c = 3.74 ; ;

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

p = a+b+c = 7.07+8.12+3.74 = 18.94 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.94 }{ 2 } = 9.47 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.47 * (9.47-7.07)(9.47-8.12)(9.47-3.74) } ; ; T = sqrt{ 174.75 } = 13.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.22 }{ 7.07 } = 3.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.22 }{ 8.12 } = 3.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.22 }{ 3.74 } = 7.07 ; ;

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{ 7.07**2-8.12**2-3.74**2 }{ 2 * 8.12 * 3.74 } ) = 60° 25'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.12**2-7.07**2-3.74**2 }{ 2 * 7.07 * 3.74 } ) = 92° 9'58" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.74**2-7.07**2-8.12**2 }{ 2 * 8.12 * 7.07 } ) = 27° 24'9" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.22 }{ 9.47 } = 1.4 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.07 }{ 2 * sin 60° 25'53" } = 4.06 ; ;




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