Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 13.45436240471   b = 10.77703296143   c = 14.03656688476

Area: T = 68
Perimeter: p = 38.2659622509
Semiperimeter: s = 19.13298112545

Angle ∠ A = α = 64.11329737337° = 64°6'47″ = 1.11989824849 rad
Angle ∠ B = β = 46.07328292758° = 46°4'22″ = 0.80441225666 rad
Angle ∠ C = γ = 69.81441969905° = 69°48'51″ = 1.21884876021 rad

Height: ha = 10.1098800389
Height: hb = 12.6277282996
Height: hc = 9.69895987984

Median: ma = 10.54875115549
Median: mb = 12.64991106407
Median: mc = 9.96224294226

Inradius: r = 3.55546613135
Circumradius: R = 7.47770879842

Vertex coordinates: A[-9; -5] B[-8; 9] C[1; -1]
Centroid: CG[-5.33333333333; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.42439752358; 3.55546613135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.8877026266° = 115°53'13″ = 1.11989824849 rad
∠ B' = β' = 133.9277170724° = 133°55'38″ = 0.80441225666 rad
∠ C' = γ' = 110.1865803009° = 110°11'9″ = 1.21884876021 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{ (-8-1)**2 + (9-(-1))**2 } ; ; a = sqrt{ 181 } = 13.45 ; ;

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{ (-9-1)**2 + (-5-(-1))**2 } ; ; b = sqrt{ 116 } = 10.77 ; ;

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{ (-9-(-8))**2 + (-5-9)**2 } ; ; c = sqrt{ 197 } = 14.04 ; ;


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 = 13.45 ; ; b = 10.77 ; ; c = 14.04 ; ;

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

p = a+b+c = 13.45+10.77+14.04 = 38.26 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 38.26 }{ 2 } = 19.13 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.13 * (19.13-13.45)(19.13-10.77)(19.13-14.04) } ; ; T = sqrt{ 4624 } = 68 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 68 }{ 13.45 } = 10.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 68 }{ 10.77 } = 12.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 68 }{ 14.04 } = 9.69 ; ;

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{ 13.45**2-10.77**2-14.04**2 }{ 2 * 10.77 * 14.04 } ) = 64° 6'47" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.77**2-13.45**2-14.04**2 }{ 2 * 13.45 * 14.04 } ) = 46° 4'22" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.04**2-13.45**2-10.77**2 }{ 2 * 10.77 * 13.45 } ) = 69° 48'51" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 68 }{ 19.13 } = 3.55 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.45 }{ 2 * sin 64° 6'47" } = 7.48 ; ;




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