Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.38551648071   b = 2.44994897428   c = 4.12331056256

Area: T = 4.82218253805
Perimeter: p = 11.95877601755
Semiperimeter: s = 5.97988800878

Angle ∠ A = α = 107.288016089° = 107°16'49″ = 1.87223920296 rad
Angle ∠ B = β = 25.7432553049° = 25°44'33″ = 0.44992923086 rad
Angle ∠ C = γ = 46.9777286061° = 46°58'38″ = 0.82199083154 rad

Height: ha = 1.79107809893
Height: hb = 3.9377003937
Height: hc = 2.33989288649

Median: ma = 2.06215528128
Median: mb = 4.63768092477
Median: mc = 3.64400549446

Inradius: r = 0.80664763484
Circumradius: R = 2.82198604404

Vertex coordinates: A[0; 1; 1] B[-2; 4; 3] C[1; 2; -1]
Centroid: CG[-0.33333333333; 2.33333333333; 1]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.721983911° = 72°43'11″ = 1.87223920296 rad
∠ B' = β' = 154.2577446951° = 154°15'27″ = 0.44992923086 rad
∠ C' = γ' = 133.0232713939° = 133°1'22″ = 0.82199083154 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{ (-2-1)**2 + (4-2)**2 + (3 - (-1))**2 } ; ; a = sqrt{ 29 } = 5.39 ; ;

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{ (0-1)**2 + (1-2)**2 + (1 - (-1))**2 } ; ; b = sqrt{ 6 } = 2.45 ; ;

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-(-2))**2 + (1-4)**2 + (1 - 3)**2 } ; ; c = sqrt{ 17 } = 4.12 ; ;


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 = 5.39 ; ; b = 2.45 ; ; c = 4.12 ; ;

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

p = a+b+c = 5.39+2.45+4.12 = 11.96 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.96 }{ 2 } = 5.98 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.98 * (5.98-5.39)(5.98-2.45)(5.98-4.12) } ; ; T = sqrt{ 23.25 } = 4.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.82 }{ 5.39 } = 1.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.82 }{ 2.45 } = 3.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.82 }{ 4.12 } = 2.34 ; ;

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{ 5.39**2-2.45**2-4.12**2 }{ 2 * 2.45 * 4.12 } ) = 107° 16'49" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.45**2-5.39**2-4.12**2 }{ 2 * 5.39 * 4.12 } ) = 25° 44'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.12**2-5.39**2-2.45**2 }{ 2 * 2.45 * 5.39 } ) = 46° 58'38" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.82 }{ 5.98 } = 0.81 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.39 }{ 2 * sin 107° 16'49" } = 2.82 ; ;




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