Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 12.36993168769   b = 2.23660679775   c = 10.77703296143

Area: T = 9
Perimeter: p = 25.37657144686
Semiperimeter: s = 12.68878572343

Angle ∠ A = α = 131.6343539337° = 131°38'1″ = 2.29774386675 rad
Angle ∠ B = β = 7.76551660184° = 7°45'55″ = 0.1365527714 rad
Angle ∠ C = γ = 40.6011294645° = 40°36'5″ = 0.70986262721 rad

Height: ha = 1.45552137502
Height: hb = 8.0549844719
Height: hc = 1.67112580436

Median: ma = 4.7176990566
Median: mb = 11.54333963806
Median: mc = 7.07110678119

Inradius: r = 0.70993396335
Circumradius: R = 8.27547943915

Vertex coordinates: A[10; -3] B[6; 7] C[9; -5]
Centroid: CG[8.33333333333; -0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[5.20218239787; 0.70993396335]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 48.36664606634° = 48°21'59″ = 2.29774386675 rad
∠ B' = β' = 172.2354833982° = 172°14'5″ = 0.1365527714 rad
∠ C' = γ' = 139.3998705355° = 139°23'55″ = 0.70986262721 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{ (6-9)**2 + (7-(-5))**2 } ; ; a = sqrt{ 153 } = 12.37 ; ;

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{ (10-9)**2 + (-3-(-5))**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (10-6)**2 + (-3-7)**2 } ; ; c = sqrt{ 116 } = 10.77 ; ;


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 = 12.37 ; ; b = 2.24 ; ; c = 10.77 ; ;

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

p = a+b+c = 12.37+2.24+10.77 = 25.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.38 }{ 2 } = 12.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.69 * (12.69-12.37)(12.69-2.24)(12.69-10.77) } ; ; T = sqrt{ 81 } = 9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9 }{ 12.37 } = 1.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9 }{ 2.24 } = 8.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9 }{ 10.77 } = 1.67 ; ;

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{ 12.37**2-2.24**2-10.77**2 }{ 2 * 2.24 * 10.77 } ) = 131° 38'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.24**2-12.37**2-10.77**2 }{ 2 * 12.37 * 10.77 } ) = 7° 45'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.77**2-12.37**2-2.24**2 }{ 2 * 2.24 * 12.37 } ) = 40° 36'5" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9 }{ 12.69 } = 0.71 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12.37 }{ 2 * sin 131° 38'1" } = 8.27 ; ;




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