Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 5.65768542495   b = 5.38551648071   c = 6.08327625303

Area: T = 14
Perimeter: p = 17.12547815869
Semiperimeter: s = 8.56223907935

Angle ∠ A = α = 58.73662683056° = 58°44'11″ = 1.02551412723 rad
Angle ∠ B = β = 54.4622322208° = 54°27'44″ = 0.95105468408 rad
Angle ∠ C = γ = 66.80114094864° = 66°48'5″ = 1.16659045405 rad

Height: ha = 4.95497474683
Height: hb = 5.1999469469
Height: hc = 4.60331716446

Median: ma = 5
Median: mb = 5.22201532545
Median: mc = 4.61097722286

Inradius: r = 1.635505735
Circumradius: R = 3.30989242348

Vertex coordinates: A[2; 3] B[1; -3] C[-3; 1]
Centroid: CG[0; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.16878981071; 1.635505735]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.2643731694° = 121°15'49″ = 1.02551412723 rad
∠ B' = β' = 125.5387677792° = 125°32'16″ = 0.95105468408 rad
∠ C' = γ' = 113.1998590514° = 113°11'55″ = 1.16659045405 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{ (1-(-3))**2 + (-3-1)**2 } ; ; a = sqrt{ 32 } = 5.66 ; ;

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-(-3))**2 + (3-1)**2 } ; ; b = sqrt{ 29 } = 5.39 ; ;

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{ (2-1)**2 + (3-(-3))**2 } ; ; c = sqrt{ 37 } = 6.08 ; ;


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 = 5.66 ; ; b = 5.39 ; ; c = 6.08 ; ;

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

p = a+b+c = 5.66+5.39+6.08 = 17.12 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.12 }{ 2 } = 8.56 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.56 * (8.56-5.66)(8.56-5.39)(8.56-6.08) } ; ; T = sqrt{ 196 } = 14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14 }{ 5.66 } = 4.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14 }{ 5.39 } = 5.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14 }{ 6.08 } = 4.6 ; ;

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{ 5.66**2-5.39**2-6.08**2 }{ 2 * 5.39 * 6.08 } ) = 58° 44'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.39**2-5.66**2-6.08**2 }{ 2 * 5.66 * 6.08 } ) = 54° 27'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.08**2-5.66**2-5.39**2 }{ 2 * 5.39 * 5.66 } ) = 66° 48'5" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14 }{ 8.56 } = 1.64 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.66 }{ 2 * sin 58° 44'11" } = 3.31 ; ;




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