Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene Pythagorean triangle.

Sides: a = 11.18803398875   b = 8.944427191   c = 6.70882039325

Area: T = 30
Perimeter: p = 26.833281573
Semiperimeter: s = 13.4166407865

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 36.87698976458° = 36°52'12″ = 0.64435011088 rad

Height: ha = 5.3676563146
Height: hb = 6.70882039325
Height: hc = 8.944427191

Median: ma = 5.59901699437
Median: mb = 8.06222577483
Median: mc = 9.55224865873

Inradius: r = 2.23660679775
Circumradius: R = 5.59901699437

Vertex coordinates: A[6; 4] B[9; 10] C[-2; 8]
Centroid: CG[4.33333333333; 7.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.67770509831; 2.23660679775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad
∠ C' = γ' = 143.1330102354° = 143°7'48″ = 0.64435011088 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{ (9-(-2))**2 + (10-8)**2 } ; ; a = sqrt{ 125 } = 11.18 ; ;

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{ (6-(-2))**2 + (4-8)**2 } ; ; b = sqrt{ 80 } = 8.94 ; ;

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{ (6-9)**2 + (4-10)**2 } ; ; c = sqrt{ 45 } = 6.71 ; ;


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 = 11.18 ; ; b = 8.94 ; ; c = 6.71 ; ;

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

p = a+b+c = 11.18+8.94+6.71 = 26.83 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.83 }{ 2 } = 13.42 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.42 * (13.42-11.18)(13.42-8.94)(13.42-6.71) } ; ; T = sqrt{ 900 } = 30 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 30 }{ 11.18 } = 5.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 30 }{ 8.94 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 30 }{ 6.71 } = 8.94 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 8.94**2+6.71**2-11.18**2 }{ 2 * 8.94 * 6.71 } ) = 90° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.18**2+6.71**2-8.94**2 }{ 2 * 11.18 * 6.71 } ) = 53° 7'48" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 11.18**2+8.94**2-6.71**2 }{ 2 * 11.18 * 8.94 } ) = 36° 52'12" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 30 }{ 13.42 } = 2.24 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.18 }{ 2 * sin 90° } = 5.59 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.94**2+2 * 6.71**2 - 11.18**2 } }{ 2 } = 5.59 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.71**2+2 * 11.18**2 - 8.94**2 } }{ 2 } = 8.062 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.94**2+2 * 11.18**2 - 6.71**2 } }{ 2 } = 9.552 ; ;
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