Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 8.06222577483   b = 11.4021754251   c = 11.18803398875

Area: T = 42.5
Perimeter: p = 30.64443518868
Semiperimeter: s = 15.32221759434

Angle ∠ A = α = 41.82201698801° = 41°49'13″ = 0.73298996582 rad
Angle ∠ B = β = 70.56599651718° = 70°33'36″ = 1.23215037123 rad
Angle ∠ C = γ = 67.6219864948° = 67°37'11″ = 1.18801892831 rad

Height: ha = 10.54329524401
Height: hb = 7.45549931641
Height: hc = 7.60326311235

Median: ma = 10.54875115549
Median: mb = 7.90656941504
Median: mc = 8.1399410298

Inradius: r = 2.77437574713
Circumradius: R = 6.04655308209

Vertex coordinates: A[-3; 2] B[7; -3] C[8; 5]
Centroid: CG[4; 1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.97989732252; 2.77437574713]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.187983012° = 138°10'47″ = 0.73298996582 rad
∠ B' = β' = 109.4440034828° = 109°26'24″ = 1.23215037123 rad
∠ C' = γ' = 112.3880135052° = 112°22'49″ = 1.18801892831 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{ (7-8)**2 + (-3-5)**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (-3-8)**2 + (2-5)**2 } ; ; b = sqrt{ 130 } = 11.4 ; ;

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 + (2-(-3))**2 } ; ; c = sqrt{ 125 } = 11.18 ; ;


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 = 8.06 ; ; b = 11.4 ; ; c = 11.18 ; ;

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

p = a+b+c = 8.06+11.4+11.18 = 30.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.64 }{ 2 } = 15.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.32 * (15.32-8.06)(15.32-11.4)(15.32-11.18) } ; ; T = sqrt{ 1806.25 } = 42.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.5 }{ 8.06 } = 10.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.5 }{ 11.4 } = 7.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.5 }{ 11.18 } = 7.6 ; ;

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{ 8.06**2-11.4**2-11.18**2 }{ 2 * 11.4 * 11.18 } ) = 41° 49'13" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 11.4**2-8.06**2-11.18**2 }{ 2 * 8.06 * 11.18 } ) = 70° 33'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.18**2-8.06**2-11.4**2 }{ 2 * 11.4 * 8.06 } ) = 67° 37'11" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.5 }{ 15.32 } = 2.77 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.06 }{ 2 * sin 41° 49'13" } = 6.05 ; ;




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