Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 4.69904157598   b = 2.44994897428   c = 4.24326406871

Area: T = 5.17220402164
Perimeter: p = 11.38325461897
Semiperimeter: s = 5.69112730949

Angle ∠ A = α = 84.47881672365° = 84°28'41″ = 1.47444221643 rad
Angle ∠ B = β = 31.31994807061° = 31°19'10″ = 0.54766280583 rad
Angle ∠ C = γ = 64.20223520573° = 64°12'8″ = 1.12105424309 rad

Height: ha = 2.20553653583
Height: hb = 4.22329531531
Height: hc = 2.43881231397

Median: ma = 2.55495097568
Median: mb = 4.30111626335
Median: mc = 3.08222070015

Inradius: r = 0.90987668313
Circumradius: R = 2.35661413093

Vertex coordinates: A[1; 3; 2] B[2; -1; 1] C[-1; 2; 3]
Centroid: CG[0.66766666667; 1.33333333333; 2]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 95.52218327635° = 95°31'19″ = 1.47444221643 rad
∠ B' = β' = 148.6810519294° = 148°40'50″ = 0.54766280583 rad
∠ C' = γ' = 115.7987647943° = 115°47'52″ = 1.12105424309 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 + (-1-2)**2 + (1 - 3)**2 } ; ; a = sqrt{ 22 } = 4.69 ; ;

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-(-1))**2 + (3-2)**2 + (2 - 3)**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{ (1-2)**2 + (3-(-1))**2 + (2 - 1)**2 } ; ; c = sqrt{ 18 } = 4.24 ; ;


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 = 4.69 ; ; b = 2.45 ; ; c = 4.24 ; ;

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

p = a+b+c = 4.69+2.45+4.24 = 11.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.38 }{ 2 } = 5.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.69 * (5.69-4.69)(5.69-2.45)(5.69-4.24) } ; ; T = sqrt{ 26.75 } = 5.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.17 }{ 4.69 } = 2.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.17 }{ 2.45 } = 4.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.17 }{ 4.24 } = 2.44 ; ;

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{ 4.69**2-2.45**2-4.24**2 }{ 2 * 2.45 * 4.24 } ) = 84° 28'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.45**2-4.69**2-4.24**2 }{ 2 * 4.69 * 4.24 } ) = 31° 19'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.24**2-4.69**2-2.45**2 }{ 2 * 2.45 * 4.69 } ) = 64° 12'8" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.17 }{ 5.69 } = 0.91 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.69 }{ 2 * sin 84° 28'41" } = 2.36 ; ;




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