Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 12.04215945788   b = 10.63301458127   c = 8.6022325267

Area: T = 44.5
Perimeter: p = 31.27440656586
Semiperimeter: s = 15.63770328293

Angle ∠ A = α = 76.72436029577° = 76°43'25″ = 1.33990794856 rad
Angle ∠ B = β = 59.22659638988° = 59°13'33″ = 1.03436880727 rad
Angle ∠ C = γ = 44.05504331436° = 44°3'2″ = 0.76988250953 rad

Height: ha = 7.3911047707
Height: hb = 8.37224157286
Height: hc = 10.34660398482

Median: ma = 7.56663729752
Median: mb = 9.01438781887
Median: mc = 10.51218980208

Inradius: r = 2.84658084399
Circumradius: R = 6.18661305421

Vertex coordinates: A[2; -3] B[7; 4] C[-5; 5]
Centroid: CG[1.33333333333; 2]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.69546949136; 2.84658084399]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.2766397042° = 103°16'35″ = 1.33990794856 rad
∠ B' = β' = 120.7744036101° = 120°46'27″ = 1.03436880727 rad
∠ C' = γ' = 135.9549566856° = 135°56'58″ = 0.76988250953 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-(-5))**2 + (4-5)**2 } ; ; a = sqrt{ 145 } = 12.04 ; ;

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-(-5))**2 + (-3-5)**2 } ; ; b = sqrt{ 113 } = 10.63 ; ;

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{ (2-7)**2 + (-3-4)**2 } ; ; c = sqrt{ 74 } = 8.6 ; ;


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 = 12.04 ; ; b = 10.63 ; ; c = 8.6 ; ;

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

p = a+b+c = 12.04+10.63+8.6 = 31.27 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.27 }{ 2 } = 15.64 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.64 * (15.64-12.04)(15.64-10.63)(15.64-8.6) } ; ; T = sqrt{ 1980.25 } = 44.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44.5 }{ 12.04 } = 7.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44.5 }{ 10.63 } = 8.37 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44.5 }{ 8.6 } = 10.35 ; ;

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{ 12.04**2-10.63**2-8.6**2 }{ 2 * 10.63 * 8.6 } ) = 76° 43'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.63**2-12.04**2-8.6**2 }{ 2 * 12.04 * 8.6 } ) = 59° 13'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.6**2-12.04**2-10.63**2 }{ 2 * 10.63 * 12.04 } ) = 44° 3'2" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44.5 }{ 15.64 } = 2.85 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.04 }{ 2 * sin 76° 43'25" } = 6.19 ; ;




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