Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 6.40331242374   b = 5.38551648071   c = 3.74216573868

Area: T = 10.06223058987
Perimeter: p = 15.53299464313
Semiperimeter: s = 7.76549732157

Angle ∠ A = α = 87.15552896121° = 87°9'19″ = 1.52111467643 rad
Angle ∠ B = β = 57.13985427737° = 57°8'19″ = 0.99772557012 rad
Angle ∠ C = γ = 35.70661676141° = 35°42'22″ = 0.62331901881 rad

Height: ha = 3.1432936331
Height: hb = 3.73770465934
Height: hc = 5.3798528742

Median: ma = 3.35441019662
Median: mb = 4.5
Median: mc = 5.61224860802

Inradius: r = 1.29658584169
Circumradius: R = 3.20655122277

Vertex coordinates: A[1; 0; 1] B[-2; 1; 3] C[4; 2; 5]
Centroid: CG[1; 1; 3]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 92.84547103879° = 92°50'41″ = 1.52111467643 rad
∠ B' = β' = 122.8611457226° = 122°51'41″ = 0.99772557012 rad
∠ C' = γ' = 144.2943832386° = 144°17'38″ = 0.62331901881 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{ (-2-4)**2 + (1-2)**2 + (3 - 5)**2 } ; ; a = sqrt{ 41 } = 6.4 ; ;

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{ (1-4)**2 + (0-2)**2 + (1 - 5)**2 } ; ; b = sqrt{ 29 } = 5.39 ; ;

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{ (1-(-2))**2 + (0-1)**2 + (1 - 3)**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 = 6.4 ; ; b = 5.39 ; ; c = 3.74 ; ;

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

p = a+b+c = 6.4+5.39+3.74 = 15.53 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.53 }{ 2 } = 7.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.76 * (7.76-6.4)(7.76-5.39)(7.76-3.74) } ; ; T = sqrt{ 101.25 } = 10.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.06 }{ 6.4 } = 3.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.06 }{ 5.39 } = 3.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.06 }{ 3.74 } = 5.38 ; ;

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{ 6.4**2-5.39**2-3.74**2 }{ 2 * 5.39 * 3.74 } ) = 87° 9'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.39**2-6.4**2-3.74**2 }{ 2 * 6.4 * 3.74 } ) = 57° 8'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.74**2-6.4**2-5.39**2 }{ 2 * 5.39 * 6.4 } ) = 35° 42'22" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.06 }{ 7.76 } = 1.3 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.4 }{ 2 * sin 87° 9'19" } = 3.21 ; ;




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