Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 12.16655250606   b = 12.53299640861   c = 10.81766538264

Area: T = 60
Perimeter: p = 35.51221429731
Semiperimeter: s = 17.75660714866

Angle ∠ A = α = 62.30105271919° = 62°18'2″ = 1.08773493252 rad
Angle ∠ B = β = 65.7722254682° = 65°46'20″ = 1.14879424007 rad
Angle ∠ C = γ = 51.9277218126° = 51°55'38″ = 0.90663009277 rad

Height: ha = 9.86439392383
Height: hb = 9.57770426136
Height: hc = 11.09440039245

Median: ma = 10
Median: mb = 9.65766039579
Median: mc = 11.10218016556

Inradius: r = 3.37991258413
Circumradius: R = 6.87700891552

Vertex coordinates: A[1; 7] B[7; -2] C[-5; -4]
Centroid: CG[1; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.52106066286; 3.37991258413]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.6999472808° = 117°41'58″ = 1.08773493252 rad
∠ B' = β' = 114.2287745318° = 114°13'40″ = 1.14879424007 rad
∠ C' = γ' = 128.0732781874° = 128°4'22″ = 0.90663009277 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 + (-2-(-4))**2 } ; ; a = sqrt{ 148 } = 12.17 ; ;

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{ (1-(-5))**2 + (7-(-4))**2 } ; ; b = sqrt{ 157 } = 12.53 ; ;

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{ (1-7)**2 + (7-(-2))**2 } ; ; c = sqrt{ 117 } = 10.82 ; ;


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 = 12.17 ; ; b = 12.53 ; ; c = 10.82 ; ;

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

p = a+b+c = 12.17+12.53+10.82 = 35.51 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35.51 }{ 2 } = 17.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.76 * (17.76-12.17)(17.76-12.53)(17.76-10.82) } ; ; T = sqrt{ 3600 } = 60 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 60 }{ 12.17 } = 9.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 60 }{ 12.53 } = 9.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 60 }{ 10.82 } = 11.09 ; ;

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{ 12.17**2-12.53**2-10.82**2 }{ 2 * 12.53 * 10.82 } ) = 62° 18'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.53**2-12.17**2-10.82**2 }{ 2 * 12.17 * 10.82 } ) = 65° 46'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.82**2-12.17**2-12.53**2 }{ 2 * 12.53 * 12.17 } ) = 51° 55'38" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 60 }{ 17.76 } = 3.38 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.17 }{ 2 * sin 62° 18'2" } = 6.87 ; ;




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