Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 13.03884048104   b = 15.65224758425   c = 8.06222577483

Area: T = 52.5
Perimeter: p = 36.75331384012
Semiperimeter: s = 18.37765692006

Angle ∠ A = α = 56.3109932474° = 56°18'36″ = 0.98327937232 rad
Angle ∠ B = β = 92.72663109939° = 92°43'35″ = 1.61883794301 rad
Angle ∠ C = γ = 30.96437565321° = 30°57'50″ = 0.54404195003 rad

Height: ha = 8.05331323829
Height: hb = 6.70882039325
Height: hc = 13.02436471319

Median: ma = 10.60766017178
Median: mb = 7.5
Median: mc = 13.82993166859

Inradius: r = 2.85768988818
Circumradius: R = 7.83551061824

Vertex coordinates: A[7; 1] B[0; 5] C[-7; -6]
Centroid: CG[0; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.13660428039; 2.85768988818]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.6990067526° = 123°41'24″ = 0.98327937232 rad
∠ B' = β' = 87.27436890061° = 87°16'25″ = 1.61883794301 rad
∠ C' = γ' = 149.0366243468° = 149°2'10″ = 0.54404195003 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{ (0-(-7))**2 + (5-(-6))**2 } ; ; a = sqrt{ 170 } = 13.04 ; ;

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{ (7-(-7))**2 + (1-(-6))**2 } ; ; b = sqrt{ 245 } = 15.65 ; ;

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{ (7-0)**2 + (1-5)**2 } ; ; c = sqrt{ 65 } = 8.06 ; ;


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 = 13.04 ; ; b = 15.65 ; ; c = 8.06 ; ;

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

p = a+b+c = 13.04+15.65+8.06 = 36.75 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 36.75 }{ 2 } = 18.38 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.38 * (18.38-13.04)(18.38-15.65)(18.38-8.06) } ; ; T = sqrt{ 2756.25 } = 52.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52.5 }{ 13.04 } = 8.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52.5 }{ 15.65 } = 6.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52.5 }{ 8.06 } = 13.02 ; ;

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{ 15.65**2+8.06**2-13.04**2 }{ 2 * 15.65 * 8.06 } ) = 56° 18'36" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13.04**2+8.06**2-15.65**2 }{ 2 * 13.04 * 8.06 } ) = 92° 43'35" ; ; gamma = 180° - alpha - beta = 180° - 56° 18'36" - 92° 43'35" = 30° 57'50" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 52.5 }{ 18.38 } = 2.86 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.04 }{ 2 * sin 56° 18'36" } = 7.84 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.65**2+2 * 8.06**2 - 13.04**2 } }{ 2 } = 10.607 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.06**2+2 * 13.04**2 - 15.65**2 } }{ 2 } = 7.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.65**2+2 * 13.04**2 - 8.06**2 } }{ 2 } = 13.829 ; ;
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