Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.22195444573   b = 6.08327625303   c = 8.944427191

Area: T = 26
Perimeter: p = 24.24765788976
Semiperimeter: s = 12.12332894488

Angle ∠ A = α = 72.89772710309° = 72°53'50″ = 1.27222973952 rad
Angle ∠ B = β = 39.09438588862° = 39°5'38″ = 0.68223165549 rad
Angle ∠ C = γ = 68.00988700828° = 68°32″ = 1.18769787035 rad

Height: ha = 5.64401919033
Height: hb = 8.54987473399
Height: hc = 5.81437767415

Median: ma = 6.10332778079
Median: mb = 8.55986213843
Median: mc = 6.40331242374

Inradius: r = 2.14546324539
Circumradius: R = 4.82330523861

Vertex coordinates: A[-1; -3] B[3; 5] C[5; -4]
Centroid: CG[2.33333333333; -0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.64395476356; 2.14546324539]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.1032728969° = 107°6'10″ = 1.27222973952 rad
∠ B' = β' = 140.9066141114° = 140°54'22″ = 0.68223165549 rad
∠ C' = γ' = 111.9911129917° = 111°59'28″ = 1.18769787035 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{ (3-5)**2 + (5-(-4))**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{ (-1-5)**2 + (-3-(-4))**2 } ; ; b = sqrt{ 37 } = 6.08 ; ;

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{ (-1-3)**2 + (-3-5)**2 } ; ; c = sqrt{ 80 } = 8.94 ; ;


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 = 6.08 ; ; c = 8.94 ; ;

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

p = a+b+c = 9.22+6.08+8.94 = 24.25 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.25 }{ 2 } = 12.12 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.12 * (12.12-9.22)(12.12-6.08)(12.12-8.94) } ; ; T = sqrt{ 676 } = 26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26 }{ 9.22 } = 5.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26 }{ 6.08 } = 8.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26 }{ 8.94 } = 5.81 ; ;

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**2-6.08**2-8.94**2 }{ 2 * 6.08 * 8.94 } ) = 72° 53'50" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.08**2-9.22**2-8.94**2 }{ 2 * 9.22 * 8.94 } ) = 39° 5'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.94**2-9.22**2-6.08**2 }{ 2 * 6.08 * 9.22 } ) = 68° 32" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26 }{ 12.12 } = 2.14 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.22 }{ 2 * sin 72° 53'50" } = 4.82 ; ;




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