Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 20.22437484162   b = 27.78548879789   c = 21.84403296678

Area: T = 219
Perimeter: p = 69.84989660629
Semiperimeter: s = 34.92444830314

Angle ∠ A = α = 46.20218330645° = 46°12'7″ = 0.80663741074 rad
Angle ∠ B = β = 82.5855369709° = 82°35'7″ = 1.44113866154 rad
Angle ∠ C = γ = 51.21327972265° = 51°12'46″ = 0.89438319308 rad

Height: ha = 21.65877061278
Height: hb = 15.76439649414
Height: hc = 20.05546423365

Median: ma = 22.85327897641
Median: mb = 15.81113883008
Median: mc = 21.70882933461

Inradius: r = 6.27106726339
Circumradius: R = 14.01095887732

Vertex coordinates: A[-2; -20] B[4; 1] C[-16; 4]
Centroid: CG[-4.66766666667; -5]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.81660464387; 6.27106726339]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.7988166936° = 133°47'53″ = 0.80663741074 rad
∠ B' = β' = 97.4154630291° = 97°24'53″ = 1.44113866154 rad
∠ C' = γ' = 128.7877202773° = 128°47'14″ = 0.89438319308 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-(-16))**2 + (1-4)**2 } ; ; a = sqrt{ 409 } = 20.22 ; ;

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{ (-2-(-16))**2 + (-20-4)**2 } ; ; b = sqrt{ 772 } = 27.78 ; ;

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{ (-2-4)**2 + (-20-1)**2 } ; ; c = sqrt{ 477 } = 21.84 ; ;


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 = 20.22 ; ; b = 27.78 ; ; c = 21.84 ; ;

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

p = a+b+c = 20.22+27.78+21.84 = 69.85 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 69.85 }{ 2 } = 34.92 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34.92 * (34.92-20.22)(34.92-27.78)(34.92-21.84) } ; ; T = sqrt{ 47961 } = 219 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 219 }{ 20.22 } = 21.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 219 }{ 27.78 } = 15.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 219 }{ 21.84 } = 20.05 ; ;

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{ 20.22**2-27.78**2-21.84**2 }{ 2 * 27.78 * 21.84 } ) = 46° 12'7" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27.78**2-20.22**2-21.84**2 }{ 2 * 20.22 * 21.84 } ) = 82° 35'7" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21.84**2-20.22**2-27.78**2 }{ 2 * 27.78 * 20.22 } ) = 51° 12'46" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 219 }{ 34.92 } = 6.27 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20.22 }{ 2 * sin 46° 12'7" } = 14.01 ; ;




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