Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 4.24326406871   b = 2.82884271247   c = 5.09990195136

Area: T = 6
Perimeter: p = 12.17700873255
Semiperimeter: s = 6.08550436627

Angle ∠ A = α = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ B = β = 33.6990067526° = 33°41'24″ = 0.58880026035 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 2.82884271247
Height: hb = 4.24326406871
Height: hc = 2.35333936217

Median: ma = 3.53655339059
Median: mb = 4.4722135955
Median: mc = 2.55495097568

Inradius: r = 0.98660241491
Circumradius: R = 2.55495097568

Vertex coordinates: A[3; 5] B[-2; 6] C[1; 3]
Centroid: CG[0.66766666667; 4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.47990362237; 0.98660241491]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ B' = β' = 146.3109932474° = 146°18'36″ = 0.58880026035 rad
∠ C' = γ' = 90° = 1.57107963268 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{ (-2-1)**2 + (6-3)**2 } ; ; a = sqrt{ 18 } = 4.24 ; ;

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{ (3-1)**2 + (5-3)**2 } ; ; b = sqrt{ 8 } = 2.83 ; ;

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{ (3-(-2))**2 + (5-6)**2 } ; ; c = sqrt{ 26 } = 5.1 ; ;


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 = 4.24 ; ; b = 2.83 ; ; c = 5.1 ; ;

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

p = a+b+c = 4.24+2.83+5.1 = 12.17 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.17 }{ 2 } = 6.09 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.09 * (6.09-4.24)(6.09-2.83)(6.09-5.1) } ; ; T = sqrt{ 36 } = 6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6 }{ 4.24 } = 2.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6 }{ 2.83 } = 4.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6 }{ 5.1 } = 2.35 ; ;

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{ 2.83**2+5.1**2-4.24**2 }{ 2 * 2.83 * 5.1 } ) = 56° 18'36" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.24**2+5.1**2-2.83**2 }{ 2 * 4.24 * 5.1 } ) = 33° 41'24" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 4.24**2+2.83**2-5.1**2 }{ 2 * 4.24 * 2.83 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6 }{ 6.09 } = 0.99 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.24 }{ 2 * sin 56° 18'36" } = 2.55 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.83**2+2 * 5.1**2 - 4.24**2 } }{ 2 } = 3.536 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 4.24**2 - 2.83**2 } }{ 2 } = 4.472 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.83**2+2 * 4.24**2 - 5.1**2 } }{ 2 } = 2.55 ; ;
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