Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 12.80662484749   b = 19.20993727123   c = 23.08767927612

Area: T = 123
Perimeter: p = 55.10224139484
Semiperimeter: s = 27.55112069742

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

Height: ha = 19.20993727123
Height: hb = 12.80662484749
Height: hc = 10.65554428129

Median: ma = 20.24884567313
Median: mb = 16.00878105936
Median: mc = 11.54333963806

Inradius: r = 4.4644414213
Circumradius: R = 11.54333963806

Vertex coordinates: A[4; 11] B[2; -12] C[-8; -4]
Centroid: CG[-0.66766666667; -1.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.9766276142; 4.4644414213]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.3109932474° = 146°18'36″ = 0.58880026035 rad
∠ B' = β' = 123.6990067526° = 123°41'24″ = 0.98327937232 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 = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 } ; ; a = sqrt{ (2-(-8))**2 + (-12-(-4))**2 } ; ; a = sqrt{ 164 } = 12.81 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 } ; ; b = sqrt{ (4-(-8))**2 + (11-(-4))**2 } ; ; b = sqrt{ 369 } = 19.21 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 } ; ; c = sqrt{ (4-2)**2 + (11-(-12))**2 } ; ; c = sqrt{ 533 } = 23.09 ; ;


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 = 12.81 ; ; b = 19.21 ; ; c = 23.09 ; ;

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

p = a+b+c = 12.81+19.21+23.09 = 55.1 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55.1 }{ 2 } = 27.55 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.55 * (27.55-12.81)(27.55-19.21)(27.55-23.09) } ; ; T = sqrt{ 15129 } = 123 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 123 }{ 12.81 } = 19.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 123 }{ 19.21 } = 12.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 123 }{ 23.09 } = 10.66 ; ;

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{ 19.21**2+23.09**2-12.81**2 }{ 2 * 19.21 * 23.09 } ) = 33° 41'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.81**2+23.09**2-19.21**2 }{ 2 * 12.81 * 23.09 } ) = 56° 18'36" ; ; gamma = 180° - alpha - beta = 180° - 33° 41'24" - 56° 18'36" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 123 }{ 27.55 } = 4.46 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.81 }{ 2 * sin 33° 41'24" } = 11.54 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.21**2+2 * 23.09**2 - 12.81**2 } }{ 2 } = 20.248 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.09**2+2 * 12.81**2 - 19.21**2 } }{ 2 } = 16.008 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.21**2+2 * 12.81**2 - 23.09**2 } }{ 2 } = 11.543 ; ;
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