Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 3.60655512755   b = 6.40331242374   c = 6.32545553203

Area: T = 11
Perimeter: p = 16.33332308332
Semiperimeter: s = 8.16766154166

Angle ∠ A = α = 32.9055242923° = 32°54'19″ = 0.57443048302 rad
Angle ∠ B = β = 74.74548812969° = 74°44'42″ = 1.30545442776 rad
Angle ∠ C = γ = 72.35498757801° = 72°21' = 1.26327435458 rad

Height: ha = 6.10217021585
Height: hb = 3.43658227615
Height: hc = 3.47985054262

Median: ma = 6.10332778079
Median: mb = 4.03111288741
Median: mc = 4.12331056256

Inradius: r = 1.34769472283
Circumradius: R = 3.31884931361

Vertex coordinates: A[0; 1] B[2; 7] C[5; 5]
Centroid: CG[2.33333333333; 4.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.36773492441; 1.34769472283]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.0954757077° = 147°5'41″ = 0.57443048302 rad
∠ B' = β' = 105.2555118703° = 105°15'18″ = 1.30545442776 rad
∠ C' = γ' = 107.655012422° = 107°39' = 1.26327435458 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{ (2-5)**2 + (7-5)**2 } ; ; a = sqrt{ 13 } = 3.61 ; ;

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{ (0-5)**2 + (1-5)**2 } ; ; b = sqrt{ 41 } = 6.4 ; ;

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{ (0-2)**2 + (1-7)**2 } ; ; c = sqrt{ 40 } = 6.32 ; ;


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 = 3.61 ; ; b = 6.4 ; ; c = 6.32 ; ;

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

p = a+b+c = 3.61+6.4+6.32 = 16.33 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.33 }{ 2 } = 8.17 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.17 * (8.17-3.61)(8.17-6.4)(8.17-6.32) } ; ; T = sqrt{ 121 } = 11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11 }{ 3.61 } = 6.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11 }{ 6.4 } = 3.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11 }{ 6.32 } = 3.48 ; ;

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{ 3.61**2-6.4**2-6.32**2 }{ 2 * 6.4 * 6.32 } ) = 32° 54'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.4**2-3.61**2-6.32**2 }{ 2 * 3.61 * 6.32 } ) = 74° 44'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.32**2-3.61**2-6.4**2 }{ 2 * 6.4 * 3.61 } ) = 72° 21' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11 }{ 8.17 } = 1.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.61 }{ 2 * sin 32° 54'19" } = 3.32 ; ;




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