Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 15.62204993518   b = 7.81102496759   c = 17.46442491966

Area: T = 61
Perimeter: p = 40.89549982243
Semiperimeter: s = 20.44774991121

Angle ∠ A = α = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ B = β = 26.56550511771° = 26°33'54″ = 0.4643647609 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 7.81102496759
Height: hb = 15.62204993518
Height: hc = 6.98656996786

Median: ma = 11.04553610172
Median: mb = 16.10112421881
Median: mc = 8.73221245983

Inradius: r = 2.98332499156
Circumradius: R = 8.73221245983

Vertex coordinates: A[-9; -5] B[8; -1] C[-4; -11]
Centroid: CG[-1.66766666667; -5.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[5.96664998311; 2.98332499156]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 153.4354948823° = 153°26'6″ = 0.4643647609 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{ (8-(-4))**2 + (-1-(-11))**2 } ; ; a = sqrt{ 244 } = 15.62 ; ;

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{ (-9-(-4))**2 + (-5-(-11))**2 } ; ; b = sqrt{ 61 } = 7.81 ; ;

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{ (-9-8)**2 + (-5-(-1))**2 } ; ; c = sqrt{ 305 } = 17.46 ; ;
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 = 15.62 ; ; b = 7.81 ; ; c = 17.46 ; ;

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

p = a+b+c = 15.62+7.81+17.46 = 40.89 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.89 }{ 2 } = 20.45 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.45 * (20.45-15.62)(20.45-7.81)(20.45-17.46) } ; ; T = sqrt{ 3721 } = 61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 61 }{ 15.62 } = 7.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 61 }{ 7.81 } = 15.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 61 }{ 17.46 } = 6.99 ; ;

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{ 7.81**2+17.46**2-15.62**2 }{ 2 * 7.81 * 17.46 } ) = 63° 26'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 15.62**2+17.46**2-7.81**2 }{ 2 * 15.62 * 17.46 } ) = 26° 33'54" ; ;
 gamma = 180° - alpha - beta = 180° - 63° 26'6" - 26° 33'54" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 61 }{ 20.45 } = 2.98 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 15.62 }{ 2 * sin 63° 26'6" } = 8.73 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 17.46**2 - 15.62**2 } }{ 2 } = 11.045 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 17.46**2+2 * 15.62**2 - 7.81**2 } }{ 2 } = 16.101 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 15.62**2 - 17.46**2 } }{ 2 } = 8.732 ; ;
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