Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 5.83109518948   b = 9.48768329805   c = 7.21111025509

Area: T = 21
Perimeter: p = 22.52988874263
Semiperimeter: s = 11.26444437131

Angle ∠ A = α = 37.87549836511° = 37°52'30″ = 0.66110431689 rad
Angle ∠ B = β = 92.72663109939° = 92°43'35″ = 1.61883794301 rad
Angle ∠ C = γ = 49.3998705355° = 49°23'55″ = 0.86221700547 rad

Height: ha = 7.2032940576
Height: hb = 4.42771887242
Height: hc = 5.82443520604

Median: ma = 7.90656941504
Median: mb = 4.52876925691
Median: mc = 7

Inradius: r = 1.86442731532
Circumradius: R = 4.74987914682

Vertex coordinates: A[-3; 6] B[-7; 12] C[-12; 9]
Centroid: CG[-7.33333333333; 9]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.08987749121; 1.86442731532]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.1255016349° = 142°7'30″ = 0.66110431689 rad
∠ B' = β' = 87.27436890061° = 87°16'25″ = 1.61883794301 rad
∠ C' = γ' = 130.6011294645° = 130°36'5″ = 0.86221700547 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{ (-7-(-12))**2 + (12-9)**2 } ; ; a = sqrt{ 34 } = 5.83 ; ;

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-(-12))**2 + (6-9)**2 } ; ; b = sqrt{ 90 } = 9.49 ; ;

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-(-7))**2 + (6-12)**2 } ; ; c = sqrt{ 52 } = 7.21 ; ;


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 = 5.83 ; ; b = 9.49 ; ; c = 7.21 ; ;

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

p = a+b+c = 5.83+9.49+7.21 = 22.53 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.53 }{ 2 } = 11.26 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.26 * (11.26-5.83)(11.26-9.49)(11.26-7.21) } ; ; T = sqrt{ 441 } = 21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 21 }{ 5.83 } = 7.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 21 }{ 9.49 } = 4.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 21 }{ 7.21 } = 5.82 ; ;

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{ 5.83**2-9.49**2-7.21**2 }{ 2 * 9.49 * 7.21 } ) = 37° 52'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9.49**2-5.83**2-7.21**2 }{ 2 * 5.83 * 7.21 } ) = 92° 43'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7.21**2-5.83**2-9.49**2 }{ 2 * 9.49 * 5.83 } ) = 49° 23'55" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 21 }{ 11.26 } = 1.86 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.83 }{ 2 * sin 37° 52'30" } = 4.75 ; ;




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