Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 21.88329842572   b = 31.28549948058   c = 31.95992693909

Area: T = 324.4687779417
Perimeter: p = 85.12772484539
Semiperimeter: s = 42.56436242269

Angle ∠ A = α = 40.46989435811° = 40°28'8″ = 0.70663163103 rad
Angle ∠ B = β = 68.10988116358° = 68°6'32″ = 1.18987230127 rad
Angle ∠ C = γ = 71.42222447831° = 71°25'20″ = 1.24765533306 rad

Height: ha = 29.65548017038
Height: hb = 20.74327094958
Height: hc = 20.30550811612

Median: ma = 29.6710804674
Median: mb = 22.48220422782
Median: mc = 21.75991182036

Inradius: r = 7.62331238601
Circumradius: R = 16.85880722083

Vertex coordinates: A[-30.75; -3.56; 64.62] B[-6.75; -19.74; 51.07] C[-5.7; 1.66; 46.62]
Centroid: CG[-14.4; -7.21333333333; 54.10333333333]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5311056419° = 139°31'52″ = 0.70663163103 rad
∠ B' = β' = 111.8911188364° = 111°53'28″ = 1.18987230127 rad
∠ C' = γ' = 108.5787755217° = 108°34'40″ = 1.24765533306 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{ (-6.75-(-5.7))**2 + (-19.74-1.66)**2 + (51.07 - 46.62)**2 } ; ; a = sqrt{ 478.865 } = 21.88 ; ;

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{ (-30.75-(-5.7))**2 + (-3.56-1.66)**2 + (64.62 - 46.62)**2 } ; ; b = sqrt{ 978.751 } = 31.28 ; ;

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{ (-30.75-(-6.75))**2 + (-3.56-(-19.74))**2 + (64.62 - 51.07)**2 } ; ; c = sqrt{ 1021.395 } = 31.96 ; ;


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 = 21.88 ; ; b = 31.28 ; ; c = 31.96 ; ;

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

p = a+b+c = 21.88+31.28+31.96 = 85.13 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 85.13 }{ 2 } = 42.56 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42.56 * (42.56-21.88)(42.56-31.28)(42.56-31.96) } ; ; T = sqrt{ 105279.34 } = 324.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 324.47 }{ 21.88 } = 29.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 324.47 }{ 31.28 } = 20.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 324.47 }{ 31.96 } = 20.31 ; ;

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{ 21.88**2-31.28**2-31.96**2 }{ 2 * 31.28 * 31.96 } ) = 40° 28'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31.28**2-21.88**2-31.96**2 }{ 2 * 21.88 * 31.96 } ) = 68° 6'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 31.96**2-21.88**2-31.28**2 }{ 2 * 31.28 * 21.88 } ) = 71° 25'20" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 324.47 }{ 42.56 } = 7.62 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21.88 }{ 2 * sin 40° 28'8" } = 16.86 ; ;




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