Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 3.60655512755   b = 4.4722135955   c = 4.12331056256

Area: T = 7
Perimeter: p = 12.20107928561
Semiperimeter: s = 6.1100396428

Angle ∠ A = α = 49.3998705355° = 49°23'55″ = 0.86221700547 rad
Angle ∠ B = β = 70.34661759419° = 70°20'46″ = 1.22877723864 rad
Angle ∠ C = γ = 60.25551187031° = 60°15'18″ = 1.05216502125 rad

Height: ha = 3.88329013736
Height: hb = 3.13304951685
Height: hc = 3.39554987505

Median: ma = 3.9055124838
Median: mb = 3.16222776602
Median: mc = 3.5

Inradius: r = 1.14774664118
Circumradius: R = 2.37443957341

Vertex coordinates: A[1; 1] B[5; 2] C[3; 5]
Centroid: CG[3; 2.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.41098094328; 1.14774664118]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.6011294645° = 130°36'5″ = 0.86221700547 rad
∠ B' = β' = 109.6543824058° = 109°39'14″ = 1.22877723864 rad
∠ C' = γ' = 119.7454881297° = 119°44'42″ = 1.05216502125 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{ (5-3)**2 + (2-5)**2 } ; ; a = sqrt{ 13 } = 3.61 ; ;

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{ (1-3)**2 + (1-5)**2 } ; ; b = sqrt{ 20 } = 4.47 ; ;

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{ (1-5)**2 + (1-2)**2 } ; ; c = sqrt{ 17 } = 4.12 ; ;


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 = 3.61 ; ; b = 4.47 ; ; c = 4.12 ; ;

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

p = a+b+c = 3.61+4.47+4.12 = 12.2 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.2 }{ 2 } = 6.1 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.1 * (6.1-3.61)(6.1-4.47)(6.1-4.12) } ; ; T = sqrt{ 49 } = 7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7 }{ 3.61 } = 3.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7 }{ 4.47 } = 3.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7 }{ 4.12 } = 3.4 ; ;

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{ 3.61**2-4.47**2-4.12**2 }{ 2 * 4.47 * 4.12 } ) = 49° 23'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.47**2-3.61**2-4.12**2 }{ 2 * 3.61 * 4.12 } ) = 70° 20'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.12**2-3.61**2-4.47**2 }{ 2 * 4.47 * 3.61 } ) = 60° 15'18" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7 }{ 6.1 } = 1.15 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.61 }{ 2 * sin 49° 23'55" } = 2.37 ; ;




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