Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.43439811321   b = 8.54440037453   c = 5.83109518948

Area: T = 24.5
Perimeter: p = 23.80989367722
Semiperimeter: s = 11.90444683861

Angle ∠ A = α = 79.59222886875° = 79°35'32″ = 1.38991474968 rad
Angle ∠ B = β = 62.96991397402° = 62°58'9″ = 1.09990188156 rad
Angle ∠ C = γ = 37.43985715723° = 37°26'19″ = 0.65334263412 rad

Height: ha = 5.1943989612
Height: hb = 5.73550162126
Height: hc = 8.4033430672

Median: ma = 5.59901699437
Median: mb = 6.5766473219
Median: mc = 8.5154693183

Inradius: r = 2.05880507424
Circumradius: R = 4.79658966565

Vertex coordinates: A[3; 2] B[8; 5] C[0; 10]
Centroid: CG[3.66766666667; 5.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.0550025889; 2.05880507424]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 100.4087711312° = 100°24'28″ = 1.38991474968 rad
∠ B' = β' = 117.031086026° = 117°1'51″ = 1.09990188156 rad
∠ C' = γ' = 142.5611428428° = 142°33'41″ = 0.65334263412 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{ (8-0)**2 + (5-10)**2 } ; ; a = sqrt{ 89 } = 9.43 ; ;

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{ (3-0)**2 + (2-10)**2 } ; ; b = sqrt{ 73 } = 8.54 ; ;

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{ (3-8)**2 + (2-5)**2 } ; ; c = sqrt{ 34 } = 5.83 ; ;


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 = 9.43 ; ; b = 8.54 ; ; c = 5.83 ; ;

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

p = a+b+c = 9.43+8.54+5.83 = 23.81 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.81 }{ 2 } = 11.9 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.9 * (11.9-9.43)(11.9-8.54)(11.9-5.83) } ; ; T = sqrt{ 600.25 } = 24.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24.5 }{ 9.43 } = 5.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24.5 }{ 8.54 } = 5.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24.5 }{ 5.83 } = 8.4 ; ;

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{ 9.43**2-8.54**2-5.83**2 }{ 2 * 8.54 * 5.83 } ) = 79° 35'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.54**2-9.43**2-5.83**2 }{ 2 * 9.43 * 5.83 } ) = 62° 58'9" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.83**2-9.43**2-8.54**2 }{ 2 * 8.54 * 9.43 } ) = 37° 26'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24.5 }{ 11.9 } = 2.06 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.43 }{ 2 * sin 79° 35'32" } = 4.8 ; ;




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