Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.21111025509   b = 14.14221356237   c = 12.80662484749

Area: T = 46
Perimeter: p = 34.15994866495
Semiperimeter: s = 17.08797433248

Angle ∠ A = α = 30.53297058999° = 30°31'47″ = 0.53328438876 rad
Angle ∠ B = β = 85.03302592719° = 85°1'49″ = 1.48440579881 rad
Angle ∠ C = γ = 64.44400348282° = 64°26'24″ = 1.12546907779 rad

Height: ha = 12.75881045132
Height: hb = 6.50553823869
Height: hc = 7.18439930469

Median: ma = 13
Median: mb = 7.61657731059
Median: mc = 9.22195444573

Inradius: r = 2.69332489046
Circumradius: R = 7.09877511814

Vertex coordinates: A[0; 5] B[-8; -5] C[-2; -9]
Centroid: CG[-3.33333333333; -3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.23441955569; 2.69332489046]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.47702941° = 149°28'13″ = 0.53328438876 rad
∠ B' = β' = 94.97697407281° = 94°58'11″ = 1.48440579881 rad
∠ C' = γ' = 115.5659965172° = 115°33'36″ = 1.12546907779 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{ (-8-(-2))**2 + (-5-(-9))**2 } ; ; a = sqrt{ 52 } = 7.21 ; ;

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{ (0-(-2))**2 + (5-(-9))**2 } ; ; b = sqrt{ 200 } = 14.14 ; ;

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{ (0-(-8))**2 + (5-(-5))**2 } ; ; c = sqrt{ 164 } = 12.81 ; ;


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 = 7.21 ; ; b = 14.14 ; ; c = 12.81 ; ;

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

p = a+b+c = 7.21+14.14+12.81 = 34.16 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 34.16 }{ 2 } = 17.08 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.08 * (17.08-7.21)(17.08-14.14)(17.08-12.81) } ; ; T = sqrt{ 2116 } = 46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46 }{ 7.21 } = 12.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46 }{ 14.14 } = 6.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46 }{ 12.81 } = 7.18 ; ;

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{ 7.21**2-14.14**2-12.81**2 }{ 2 * 14.14 * 12.81 } ) = 30° 31'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 14.14**2-7.21**2-12.81**2 }{ 2 * 7.21 * 12.81 } ) = 85° 1'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.81**2-7.21**2-14.14**2 }{ 2 * 14.14 * 7.21 } ) = 64° 26'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46 }{ 17.08 } = 2.69 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.21 }{ 2 * sin 30° 31'47" } = 7.1 ; ;




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