Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 4.24326406871   b = 2.23660679775   c = 4.12331056256

Area: T = 4.5
Perimeter: p = 10.60218142902
Semiperimeter: s = 5.30109071451

Angle ∠ A = α = 77.47111922908° = 77°28'16″ = 1.35221273809 rad
Angle ∠ B = β = 30.96437565321° = 30°57'50″ = 0.54404195003 rad
Angle ∠ C = γ = 71.56550511771° = 71°33'54″ = 1.24990457724 rad

Height: ha = 2.12113203436
Height: hb = 4.02549223595
Height: hc = 2.18328206253

Median: ma = 2.55495097568
Median: mb = 4.03111288741
Median: mc = 2.69325824036

Inradius: r = 0.84989113046
Circumradius: R = 2.17330674684

Vertex coordinates: A[1; 4] B[5; 5] C[2; 2]
Centroid: CG[2.66766666667; 3.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.41548521743; 0.84989113046]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102.5298807709° = 102°31'44″ = 1.35221273809 rad
∠ B' = β' = 149.0366243468° = 149°2'10″ = 0.54404195003 rad
∠ C' = γ' = 108.4354948823° = 108°26'6″ = 1.24990457724 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 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (5-2)**2 + (5-2)**2 } ; ; a = sqrt{ 18 } = 4.24 ; ;

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 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (1-2)**2 + (4-2)**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (1-5)**2 + (4-5)**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 = 4.24 ; ; b = 2.24 ; ; c = 4.12 ; ;

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

p = a+b+c = 4.24+2.24+4.12 = 10.6 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.6 }{ 2 } = 5.3 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.3 * (5.3-4.24)(5.3-2.24)(5.3-4.12) } ; ; T = sqrt{ 20.25 } = 4.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.5 }{ 4.24 } = 2.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.5 }{ 2.24 } = 4.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.5 }{ 4.12 } = 2.18 ; ;

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{ 4.24**2-2.24**2-4.12**2 }{ 2 * 2.24 * 4.12 } ) = 77° 28'16" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.24**2-4.24**2-4.12**2 }{ 2 * 4.24 * 4.12 } ) = 30° 57'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.12**2-4.24**2-2.24**2 }{ 2 * 2.24 * 4.24 } ) = 71° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.5 }{ 5.3 } = 0.85 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.24 }{ 2 * sin 77° 28'16" } = 2.17 ; ;




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