Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.944427191   b = 15.23215462117   c = 11.66219037897

Area: T = 52
Perimeter: p = 35.83877219114
Semiperimeter: s = 17.91988609557

Angle ∠ A = α = 35.83876529543° = 35°50'16″ = 0.62554850402 rad
Angle ∠ B = β = 94.3998705355° = 94°23'55″ = 1.64875682181 rad
Angle ∠ C = γ = 49.76436416907° = 49°45'49″ = 0.86985393953 rad

Height: ha = 11.6287553483
Height: hb = 6.82879345087
Height: hc = 8.91879264274

Median: ma = 12.80662484749
Median: mb = 7.07110678119
Median: mc = 11.04553610172

Inradius: r = 2.90219701715
Circumradius: R = 7.6388271746

Vertex coordinates: A[5; 10] B[-1; 0] C[-9; 4]
Centroid: CG[-1.66766666667; 4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.22332284747; 2.90219701715]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.1622347046° = 144°9'44″ = 0.62554850402 rad
∠ B' = β' = 85.6011294645° = 85°36'5″ = 1.64875682181 rad
∠ C' = γ' = 130.2366358309° = 130°14'11″ = 0.86985393953 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{ (-1-(-9))**2 + (0-4)**2 } ; ; a = sqrt{ 80 } = 8.94 ; ;

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{ (5-(-9))**2 + (10-4)**2 } ; ; b = sqrt{ 232 } = 15.23 ; ;

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{ (5-(-1))**2 + (10-0)**2 } ; ; c = sqrt{ 136 } = 11.66 ; ;


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 = 8.94 ; ; b = 15.23 ; ; c = 11.66 ; ;

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

p = a+b+c = 8.94+15.23+11.66 = 35.84 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 35.84 }{ 2 } = 17.92 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17.92 * (17.92-8.94)(17.92-15.23)(17.92-11.66) } ; ; T = sqrt{ 2704 } = 52 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52 }{ 8.94 } = 11.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52 }{ 15.23 } = 6.83 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52 }{ 11.66 } = 8.92 ; ;

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{ 15.23**2+11.66**2-8.94**2 }{ 2 * 15.23 * 11.66 } ) = 35° 50'16" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.94**2+11.66**2-15.23**2 }{ 2 * 8.94 * 11.66 } ) = 94° 23'55" ; ; gamma = 180° - alpha - beta = 180° - 35° 50'16" - 94° 23'55" = 49° 45'49" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 52 }{ 17.92 } = 2.9 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.94 }{ 2 * sin 35° 50'16" } = 7.64 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.23**2+2 * 11.66**2 - 8.94**2 } }{ 2 } = 12.806 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.66**2+2 * 8.94**2 - 15.23**2 } }{ 2 } = 7.071 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.23**2+2 * 8.94**2 - 11.66**2 } }{ 2 } = 11.045 ; ;
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