Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 336.0065952328   b = 270   c = 200

Area: T = 27000
Perimeter: p = 806.0065952328
Semiperimeter: s = 403.0032976164

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 53.4711144633° = 53°28'16″ = 0.93332475287 rad
Angle ∠ C = γ = 36.5298855367° = 36°31'44″ = 0.63875487981 rad

Height: ha = 160.7111438669
Height: hb = 200
Height: hc = 270

Median: ma = 168.0032976164
Median: mb = 241.299857024
Median: mc = 287.9243600978

Inradius: r = 66.99770238359
Circumradius: R = 168.0032976164

Vertex coordinates: A[525; 260] B[725; 260] C[525; -10]
Centroid: CG[591.6676666667; 170]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[49.62774250636; 66.99770238359]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 126.5298855367° = 126°31'44″ = 0.93332475287 rad
∠ C' = γ' = 143.4711144633° = 143°28'16″ = 0.63875487981 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{ (725-525)**2 + (260-(-10))**2 } ; ; a = sqrt{ 112900 } = 336.01 ; ;

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{ (525-525)**2 + (260-(-10))**2 } ; ; b = sqrt{ 72900 } = 270 ; ;

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{ (525-725)**2 + (260-260)**2 } ; ; c = sqrt{ 40000 } = 200 ; ;


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 = 336.01 ; ; b = 270 ; ; c = 200 ; ;

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

p = a+b+c = 336.01+270+200 = 806.01 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 806.01 }{ 2 } = 403 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 403 * (403-336.01)(403-270)(403-200) } ; ; T = sqrt{ 729000000 } = 27000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27000 }{ 336.01 } = 160.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27000 }{ 270 } = 200 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27000 }{ 200 } = 270 ; ;

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{ 336.01**2-270**2-200**2 }{ 2 * 270 * 200 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 270**2-336.01**2-200**2 }{ 2 * 336.01 * 200 } ) = 53° 28'16" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200**2-336.01**2-270**2 }{ 2 * 270 * 336.01 } ) = 36° 31'44" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27000 }{ 403 } = 67 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 336.01 }{ 2 * sin 90° } = 168 ; ;




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