Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 6.70882039325   b = 6.32545553203   c = 12.04215945788

Area: T = 15
Perimeter: p = 25.07443538316
Semiperimeter: s = 12.53771769158

Angle ∠ A = α = 23.19985905136° = 23°11'55″ = 0.40548917863 rad
Angle ∠ B = β = 21.80114094864° = 21°48'5″ = 0.38105063771 rad
Angle ∠ C = γ = 135° = 2.35661944902 rad

Height: ha = 4.4722135955
Height: hb = 4.74334164903
Height: hc = 2.49113643956

Median: ma = 9.01438781887
Median: mb = 9.22195444573
Median: mc = 2.5

Inradius: r = 1.19664415993
Circumradius: R = 8.5154693183

Vertex coordinates: A[-2; 6] B[6; -3] C[0; 0]
Centroid: CG[1.33333333333; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.99111039983; 1.19664415993]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 156.8011409486° = 156°48'5″ = 0.40548917863 rad
∠ B' = β' = 158.1998590514° = 158°11'55″ = 0.38105063771 rad
∠ C' = γ' = 45° = 2.35661944902 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{ (6-0)**2 + (-3-0)**2 } ; ; a = sqrt{ 45 } = 6.71 ; ;

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{ (-2-0)**2 + (6-0)**2 } ; ; b = sqrt{ 40 } = 6.32 ; ;

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{ (-2-6)**2 + (6-(-3))**2 } ; ; c = sqrt{ 145 } = 12.04 ; ;


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 = 6.71 ; ; b = 6.32 ; ; c = 12.04 ; ;

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

p = a+b+c = 6.71+6.32+12.04 = 25.07 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.07 }{ 2 } = 12.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.54 * (12.54-6.71)(12.54-6.32)(12.54-12.04) } ; ; T = sqrt{ 225 } = 15 ; ;

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 }{ 6.71 } = 4.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15 }{ 6.32 } = 4.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15 }{ 12.04 } = 2.49 ; ;

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{ 6.71**2-6.32**2-12.04**2 }{ 2 * 6.32 * 12.04 } ) = 23° 11'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.32**2-6.71**2-12.04**2 }{ 2 * 6.71 * 12.04 } ) = 21° 48'5" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.04**2-6.71**2-6.32**2 }{ 2 * 6.32 * 6.71 } ) = 135° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15 }{ 12.54 } = 1.2 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.71 }{ 2 * sin 23° 11'55" } = 8.51 ; ;




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