Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 3.60655512755   b = 5.38551648071   c = 5.83109518948

Area: T = 9.5
Perimeter: p = 14.82216679774
Semiperimeter: s = 7.41108339887

Angle ∠ A = α = 37.23548339816° = 37°14'5″ = 0.65498704494 rad
Angle ∠ B = β = 64.65438240581° = 64°39'14″ = 1.12884221038 rad
Angle ∠ C = γ = 78.11113419604° = 78°6'41″ = 1.36333001004 rad

Height: ha = 5.27696518641
Height: hb = 3.52882114254
Height: hc = 3.25884731177

Median: ma = 5.31550729064
Median: mb = 4.03111288741
Median: mc = 3.53655339059

Inradius: r = 1.28219070046
Circumradius: R = 2.9799384383

Vertex coordinates: A[0; 1] B[5; 4] C[2; 6]
Centroid: CG[2.33333333333; 3.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.60772191074; 1.28219070046]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.76551660184° = 142°45'55″ = 0.65498704494 rad
∠ B' = β' = 115.3466175942° = 115°20'46″ = 1.12884221038 rad
∠ C' = γ' = 101.88986580396° = 101°53'19″ = 1.36333001004 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{ (5-2)**2 + (4-6)**2 } ; ; a = sqrt{ 13 } = 3.61 ; ;

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{ (0-2)**2 + (1-6)**2 } ; ; b = sqrt{ 29 } = 5.39 ; ;

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{ (0-5)**2 + (1-4)**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 = 3.61 ; ; b = 5.39 ; ; c = 5.83 ; ;

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

p = a+b+c = 3.61+5.39+5.83 = 14.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.82 }{ 2 } = 7.41 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.41 * (7.41-3.61)(7.41-5.39)(7.41-5.83) } ; ; T = sqrt{ 90.25 } = 9.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.5 }{ 3.61 } = 5.27 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.5 }{ 5.39 } = 3.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.5 }{ 5.83 } = 3.26 ; ;

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{ 5.39**2+5.83**2-3.61**2 }{ 2 * 5.39 * 5.83 } ) = 37° 14'5" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.61**2+5.83**2-5.39**2 }{ 2 * 3.61 * 5.83 } ) = 64° 39'14" ; ; gamma = 180° - alpha - beta = 180° - 37° 14'5" - 64° 39'14" = 78° 6'41" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.5 }{ 7.41 } = 1.28 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3.61 }{ 2 * sin 37° 14'5" } = 2.98 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 5.83**2 - 3.61**2 } }{ 2 } = 5.315 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.83**2+2 * 3.61**2 - 5.39**2 } }{ 2 } = 4.031 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 3.61**2 - 5.83**2 } }{ 2 } = 3.536 ; ;
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