Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 13.89224439894   b = 11.18803398875   c = 12.16655250606

Area: T = 65
Perimeter: p = 37.23883089375
Semiperimeter: s = 18.61991544688

Angle ∠ A = α = 72.89772710309° = 72°53'50″ = 1.27222973952 rad
Angle ∠ B = β = 50.28112406284° = 50°16'52″ = 0.8787573201 rad
Angle ∠ C = γ = 56.82114883406° = 56°49'17″ = 0.99217220574 rad

Height: ha = 9.35876047597
Height: hb = 11.6287553483
Height: hc = 10.68659341748

Median: ma = 9.3944147114
Median: mb = 11.88004237212
Median: mc = 11.04553610172

Inradius: r = 3.49110285593
Circumradius: R = 7.26876025853

Vertex coordinates: A[-8; -2] B[4; -4] C[-3; 8]
Centroid: CG[-2.33333333333; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.99002391108; 3.49110285593]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.1032728969° = 107°6'10″ = 1.27222973952 rad
∠ B' = β' = 129.7198759372° = 129°43'8″ = 0.8787573201 rad
∠ C' = γ' = 123.1798511659° = 123°10'43″ = 0.99217220574 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{ (4-(-3))**2 + (-4-8)**2 } ; ; a = sqrt{ 193 } = 13.89 ; ;

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{ (-8-(-3))**2 + (-2-8)**2 } ; ; b = sqrt{ 125 } = 11.18 ; ;

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{ (-8-4)**2 + (-2-(-4))**2 } ; ; c = sqrt{ 148 } = 12.17 ; ;


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 = 13.89 ; ; b = 11.18 ; ; c = 12.17 ; ;

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

p = a+b+c = 13.89+11.18+12.17 = 37.24 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.24 }{ 2 } = 18.62 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.62 * (18.62-13.89)(18.62-11.18)(18.62-12.17) } ; ; T = sqrt{ 4225 } = 65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 65 }{ 13.89 } = 9.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 65 }{ 11.18 } = 11.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 65 }{ 12.17 } = 10.69 ; ;

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{ 13.89**2-11.18**2-12.17**2 }{ 2 * 11.18 * 12.17 } ) = 72° 53'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.18**2-13.89**2-12.17**2 }{ 2 * 13.89 * 12.17 } ) = 50° 16'52" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.17**2-13.89**2-11.18**2 }{ 2 * 11.18 * 13.89 } ) = 56° 49'17" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 65 }{ 18.62 } = 3.49 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.89 }{ 2 * sin 72° 53'50" } = 7.27 ; ;




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