Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 4.89989794856   b = 9.43439811321   c = 4.5832575695

Area: T = 2.23660679775
Perimeter: p = 18.91655363126
Semiperimeter: s = 9.45877681563

Angle ∠ A = α = 5.93875953156° = 5°56'15″ = 0.10436305879 rad
Angle ∠ B = β = 168.5109540096° = 168°30'34″ = 2.94110462957 rad
Angle ∠ C = γ = 5.55328645881° = 5°33'10″ = 0.097691577 rad

Height: ha = 0.91328709292
Height: hb = 0.47440454631
Height: hc = 0.97659000729

Median: ma = 7
Median: mb = 0.5
Median: mc = 7.15989105316

Inradius: r = 0.23664266009
Circumradius: R = 23.67991047128

Vertex coordinates: A[2; -3; 5] B[0; 1; 4] C[-2; 5; 2]
Centroid: CG[0; 1; 3.66766666667]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 174.0622404684° = 174°3'45″ = 0.10436305879 rad
∠ B' = β' = 11.49904599037° = 11°29'26″ = 2.94110462957 rad
∠ C' = γ' = 174.4477135412° = 174°26'50″ = 0.097691577 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-(-2))**2 + (1-5)**2 + (4 - 2)**2 } ; ; a = sqrt{ 24 } = 4.9 ; ;

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-(-2))**2 + (-3-5)**2 + (5 - 2)**2 } ; ; b = sqrt{ 89 } = 9.43 ; ;

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{ (2-0)**2 + (-3-1)**2 + (5 - 4)**2 } ; ; c = sqrt{ 21 } = 4.58 ; ;


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 = 4.9 ; ; b = 9.43 ; ; c = 4.58 ; ;

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

p = a+b+c = 4.9+9.43+4.58 = 18.92 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18.92 }{ 2 } = 9.46 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.46 * (9.46-4.9)(9.46-9.43)(9.46-4.58) } ; ; T = sqrt{ 5 } = 2.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.24 }{ 4.9 } = 0.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.24 }{ 9.43 } = 0.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.24 }{ 4.58 } = 0.98 ; ;

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{ 4.9**2-9.43**2-4.58**2 }{ 2 * 9.43 * 4.58 } ) = 5° 56'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.43**2-4.9**2-4.58**2 }{ 2 * 4.9 * 4.58 } ) = 168° 30'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.58**2-4.9**2-9.43**2 }{ 2 * 9.43 * 4.9 } ) = 5° 33'10" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.24 }{ 9.46 } = 0.24 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.9 }{ 2 * sin 5° 56'15" } = 23.68 ; ;




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