Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 16.76330546142   b = 7.61657731059   c = 19.10549731745

Area: T = 63.5
Perimeter: p = 43.48438008946
Semiperimeter: s = 21.74219004473

Angle ∠ A = α = 60.79224035289° = 60°47'33″ = 1.06110276018 rad
Angle ∠ B = β = 23.36330305938° = 23°21'47″ = 0.40877618071 rad
Angle ∠ C = γ = 95.84545658774° = 95°50'40″ = 1.67328032447 rad

Height: ha = 7.57661848257
Height: hb = 16.67659169732
Height: hc = 6.64774838169

Median: ma = 11.8854864324
Median: mb = 17.56441680703
Median: mc = 8.84659030065

Inradius: r = 2.92106278519
Circumradius: R = 9.60224017523

Vertex coordinates: A[-10; -3] B[9; -1] C[-7; 4]
Centroid: CG[-2.66766666667; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[6.76111384918; 2.92106278519]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119.2087596471° = 119°12'27″ = 1.06110276018 rad
∠ B' = β' = 156.6376969406° = 156°38'13″ = 0.40877618071 rad
∠ C' = γ' = 84.15554341226° = 84°9'20″ = 1.67328032447 rad

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How did we calculate this triangle?

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

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (9-(-7))**2 + (-1-4)**2 } ; ; a = sqrt{ 281 } = 16.76 ; ;

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

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (-10-(-7))**2 + (-3-4)**2 } ; ; b = sqrt{ 58 } = 7.62 ; ;

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

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (-10-9)**2 + (-3-(-1))**2 } ; ; c = sqrt{ 365 } = 19.1 ; ;


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 = 16.76 ; ; b = 7.62 ; ; c = 19.1 ; ;

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

p = a+b+c = 16.76+7.62+19.1 = 43.48 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43.48 }{ 2 } = 21.74 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.74 * (21.74-16.76)(21.74-7.62)(21.74-19.1) } ; ; T = sqrt{ 4032.25 } = 63.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 63.5 }{ 16.76 } = 7.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 63.5 }{ 7.62 } = 16.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 63.5 }{ 19.1 } = 6.65 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 16.76**2-7.62**2-19.1**2 }{ 2 * 7.62 * 19.1 } ) = 60° 47'33" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.62**2-16.76**2-19.1**2 }{ 2 * 16.76 * 19.1 } ) = 23° 21'47" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 19.1**2-16.76**2-7.62**2 }{ 2 * 7.62 * 16.76 } ) = 95° 50'40" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 63.5 }{ 21.74 } = 2.92 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.76 }{ 2 * sin 60° 47'33" } = 9.6 ; ;




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