Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 18.78882942281   b = 7.61657731059   c = 14.03656688476

Area: T = 47.5
Perimeter: p = 40.44397361815
Semiperimeter: s = 20.22198680908

Angle ∠ A = α = 117.2844207294° = 117°17'3″ = 2.04769955779 rad
Angle ∠ B = β = 21.11655068655° = 21°6'56″ = 0.3698535118 rad
Angle ∠ C = γ = 41.66002858409° = 41°36'1″ = 0.72660619577 rad

Height: ha = 5.05663398064
Height: hb = 12.47441111217
Height: hc = 6.76884697489

Median: ma = 6.26549820431
Median: mb = 16.14400123916
Median: mc = 12.5

Inradius: r = 2.34991745736
Circumradius: R = 10.57701429715

Vertex coordinates: A[1; -5] B[2; 9] C[-6; -8]
Centroid: CG[-1; -1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[6.08331257379; 2.34991745736]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 62.71657927064° = 62°42'57″ = 2.04769955779 rad
∠ B' = β' = 158.8844493134° = 158°53'4″ = 0.3698535118 rad
∠ C' = γ' = 138.4399714159° = 138°23'59″ = 0.72660619577 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{ (2-(-6))**2 + (9-(-8))**2 } ; ; a = sqrt{ 353 } = 18.79 ; ;

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-(-6))**2 + (-5-(-8))**2 } ; ; b = sqrt{ 58 } = 7.62 ; ;

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{ (1-2)**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 = 18.79 ; ; b = 7.62 ; ; c = 14.04 ; ;

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

p = a+b+c = 18.79+7.62+14.04 = 40.44 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.44 }{ 2 } = 20.22 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.22 * (20.22-18.79)(20.22-7.62)(20.22-14.04) } ; ; T = sqrt{ 2256.25 } = 47.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 47.5 }{ 18.79 } = 5.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 47.5 }{ 7.62 } = 12.47 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 47.5 }{ 14.04 } = 6.77 ; ;

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{ 18.79**2-7.62**2-14.04**2 }{ 2 * 7.62 * 14.04 } ) = 117° 17'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.62**2-18.79**2-14.04**2 }{ 2 * 18.79 * 14.04 } ) = 21° 6'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.04**2-18.79**2-7.62**2 }{ 2 * 7.62 * 18.79 } ) = 41° 36'1" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 47.5 }{ 20.22 } = 2.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18.79 }{ 2 * sin 117° 17'3" } = 10.57 ; ;




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