Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.22195444573   b = 3.60655512755   c = 8.6022325267

Area: T = 15.5
Perimeter: p = 21.42774209998
Semiperimeter: s = 10.71437104999

Angle ∠ A = α = 88.1522389734° = 88°9'9″ = 1.53985494444 rad
Angle ∠ B = β = 23.00988700828° = 23°32″ = 0.40215805401 rad
Angle ∠ C = γ = 68.83987401832° = 68°50'19″ = 1.20114626691 rad

Height: ha = 3.36224220962
Height: hb = 8.59878530415
Height: hc = 3.60436768011

Median: ma = 4.7176990566
Median: mb = 8.73221245983
Median: mc = 5.52326805086

Inradius: r = 1.44767443376
Circumradius: R = 4.61221700297

Vertex coordinates: A[-2; 2] B[5; -3] C[-4; -1]
Centroid: CG[-0.33333333333; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.40768495691; 1.44767443376]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 91.8487610266° = 91°50'51″ = 1.53985494444 rad
∠ B' = β' = 156.9911129917° = 156°59'28″ = 0.40215805401 rad
∠ C' = γ' = 111.1611259817° = 111°9'41″ = 1.20114626691 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{ (5-(-4))**2 + (-3-(-1))**2 } ; ; a = sqrt{ 85 } = 9.22 ; ;

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-(-4))**2 + (2-(-1))**2 } ; ; b = sqrt{ 13 } = 3.61 ; ;

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{ (-2-5)**2 + (2-(-3))**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 = 9.22 ; ; b = 3.61 ; ; c = 8.6 ; ;

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

p = a+b+c = 9.22+3.61+8.6 = 21.43 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.43 }{ 2 } = 10.71 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.71 * (10.71-9.22)(10.71-3.61)(10.71-8.6) } ; ; T = sqrt{ 240.25 } = 15.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.5 }{ 9.22 } = 3.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.5 }{ 3.61 } = 8.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.5 }{ 8.6 } = 3.6 ; ;

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{ 9.22**2-3.61**2-8.6**2 }{ 2 * 3.61 * 8.6 } ) = 88° 9'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3.61**2-9.22**2-8.6**2 }{ 2 * 9.22 * 8.6 } ) = 23° 32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.6**2-9.22**2-3.61**2 }{ 2 * 3.61 * 9.22 } ) = 68° 50'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.5 }{ 10.71 } = 1.45 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.22 }{ 2 * sin 88° 9'9" } = 4.61 ; ;




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