Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 4.4722135955   b = 6.70882039325   c = 5.38551648071

Area: T = 12
Perimeter: p = 16.56655046946
Semiperimeter: s = 8.28327523473

Angle ∠ A = α = 41.63435393366° = 41°38'1″ = 0.72766423407 rad
Angle ∠ B = β = 85.23663583093° = 85°14'11″ = 1.48876550949 rad
Angle ∠ C = γ = 53.13301023542° = 53°7'48″ = 0.9277295218 rad

Height: ha = 5.3676563146
Height: hb = 3.5787708764
Height: hc = 4.45766881162

Median: ma = 5.65768542495
Median: mb = 3.64400549446
Median: mc = 5.02549378106

Inradius: r = 1.44987937701
Circumradius: R = 3.36657280045

Vertex coordinates: A[-3; 0] B[2; 2] C[0; 6]
Centroid: CG[-0.33333333333; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.12107328142; 1.44987937701]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.3666460663° = 138°21'59″ = 0.72766423407 rad
∠ B' = β' = 94.76436416907° = 94°45'49″ = 1.48876550949 rad
∠ C' = γ' = 126.8769897646° = 126°52'12″ = 0.9277295218 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{ (2-0)**2 + (2-6)**2 } ; ; a = sqrt{ 20 } = 4.47 ; ;

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{ (-3-0)**2 + (0-6)**2 } ; ; b = sqrt{ 45 } = 6.71 ; ;

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{ (-3-2)**2 + (0-2)**2 } ; ; c = sqrt{ 29 } = 5.39 ; ;


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 = 4.47 ; ; b = 6.71 ; ; c = 5.39 ; ;

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

p = a+b+c = 4.47+6.71+5.39 = 16.57 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.57 }{ 2 } = 8.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.28 * (8.28-4.47)(8.28-6.71)(8.28-5.39) } ; ; T = sqrt{ 144 } = 12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12 }{ 4.47 } = 5.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12 }{ 6.71 } = 3.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12 }{ 5.39 } = 4.46 ; ;

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{ 4.47**2-6.71**2-5.39**2 }{ 2 * 6.71 * 5.39 } ) = 41° 38'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.71**2-4.47**2-5.39**2 }{ 2 * 4.47 * 5.39 } ) = 85° 14'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.39**2-4.47**2-6.71**2 }{ 2 * 6.71 * 4.47 } ) = 53° 7'48" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12 }{ 8.28 } = 1.45 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.47 }{ 2 * sin 41° 38'1" } = 3.37 ; ;




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