Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 7.28801098893   b = 15.29770585408   c = 8.06222577483

Area: T = 4.5
Perimeter: p = 30.63994261784
Semiperimeter: s = 15.32197130892

Angle ∠ A = α = 4.18549161251° = 4°11'6″ = 0.07330405653 rad
Angle ∠ B = β = 171.1879620448° = 171°10'47″ = 2.98876479891 rad
Angle ∠ C = γ = 4.63554634269° = 4°38'8″ = 0.08109040992 rad

Height: ha = 1.23662450755
Height: hb = 0.58883484054
Height: hc = 1.11663126113

Median: ma = 11.67326175299
Median: mb = 0.70771067812
Median: mc = 11.28105141727

Inradius: r = 0.29437391826
Circumradius: R = 49.8880412544

Vertex coordinates: A[0; -9] B[1; -1] C[3; 6]
Centroid: CG[1.33333333333; -1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-1.89329858432; 0.29437391826]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 175.81550838749° = 175°48'54″ = 0.07330405653 rad
∠ B' = β' = 8.8220379552° = 8°49'13″ = 2.98876479891 rad
∠ C' = γ' = 175.36545365731° = 175°21'52″ = 0.08109040992 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = |BC| = | beta - gamma | ; ; 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{ (1-3)**2 + (-1-6)**2 } ; ; a = sqrt{ 53 } = 7.28 ; ;

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

b = |AC| = | alpha - gamma | ; ; 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{ (0-3)**2 + (-9-6)**2 } ; ; b = sqrt{ 234 } = 15.3 ; ;

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

c = |AB| = | alpha - beta | ; ; 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{ (0-1)**2 + (-9-(-1))**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 = 7.28 ; ; b = 15.3 ; ; c = 8.06 ; ;

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

p = a+b+c = 7.28+15.3+8.06 = 30.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.64 }{ 2 } = 15.32 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.32 * (15.32-7.28)(15.32-15.3)(15.32-8.06) } ; ; T = sqrt{ 20.25 } = 4.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.5 }{ 7.28 } = 1.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.5 }{ 15.3 } = 0.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.5 }{ 8.06 } = 1.12 ; ;

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{ 15.3**2+8.06**2-7.28**2 }{ 2 * 15.3 * 8.06 } ) = 4° 11'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.28**2+8.06**2-15.3**2 }{ 2 * 7.28 * 8.06 } ) = 171° 10'47" ; ; gamma = 180° - alpha - beta = 180° - 4° 11'6" - 171° 10'47" = 4° 38'8" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.5 }{ 15.32 } = 0.29 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.28 }{ 2 * sin 4° 11'6" } = 49.88 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.3**2+2 * 8.06**2 - 7.28**2 } }{ 2 } = 11.673 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.06**2+2 * 7.28**2 - 15.3**2 } }{ 2 } = 0.707 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15.3**2+2 * 7.28**2 - 8.06**2 } }{ 2 } = 11.281 ; ;
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