Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 0.34114   b = 0.36766464919   c = 0.13437

Area: T = 0.023282259
Perimeter: p = 0.84217464919
Semiperimeter: s = 0.42108732459

Angle ∠ A = α = 68.61435597808° = 68°36'49″ = 1.19875325297 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 21.38664402192° = 21°23'11″ = 0.37332637971 rad

Height: ha = 0.13437
Height: hb = 0.12444937045
Height: hc = 0.34114

Median: ma = 0.21768275352
Median: mb = 0.18333232459
Median: mc = 0.34878834323

Inradius: r = 0.05442267541
Circumradius: R = 0.18333232459

Vertex coordinates: A[0.3506; -0.34; 0.495] B[0.3506; -0.4737; 0.495] C[0.3506; -0.4737; 0.83664]
Centroid: CG[0.35106; -0.42991333333; 0.60988]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.3866440219° = 111°23'11″ = 1.19875325297 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 158.6143559781° = 158°36'49″ = 0.37332637971 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{ (0.351-0.351)**2 + (-0.474-(-0.474))**2 + (0.495 - 0.836)**2 } ; ; a = sqrt{ 0.117 } = 0.34 ; ;

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{ (0.351-0.351)**2 + (-0.34-(-0.474))**2 + (0.495 - 0.836)**2 } ; ; b = sqrt{ 0.134 } = 0.37 ; ;

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{ (0.351-0.351)**2 + (-0.34-(-0.474))**2 + (0.495 - 0.495)**2 } ; ; c = sqrt{ 0.018 } = 0.13 ; ;


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 = 0.34 ; ; b = 0.37 ; ; c = 0.13 ; ;

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

p = a+b+c = 0.34+0.37+0.13 = 0.84 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 0.84 }{ 2 } = 0.42 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.42 * (0.42-0.34)(0.42-0.37)(0.42-0.13) } ; ; T = sqrt{ 0 } = 0.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.02 }{ 0.34 } = 0.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.02 }{ 0.37 } = 0.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.02 }{ 0.13 } = 0.34 ; ;

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{ 0.34**2-0.37**2-0.13**2 }{ 2 * 0.37 * 0.13 } ) = 68° 36'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.37**2-0.34**2-0.13**2 }{ 2 * 0.34 * 0.13 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.13**2-0.34**2-0.37**2 }{ 2 * 0.37 * 0.34 } ) = 21° 23'11" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.02 }{ 0.42 } = 0.05 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 0.34 }{ 2 * sin 68° 36'49" } = 0.18 ; ;




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