Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 46.3989654019   b = 49.09217508345   c = 9.48768329805

Area: T = 216
Perimeter: p = 104.9688237834
Semiperimeter: s = 52.4844118917

Angle ∠ A = α = 68.06215195323° = 68°3'41″ = 1.18878976097 rad
Angle ∠ B = β = 101.0043540852° = 101°13″ = 1.7632844344 rad
Angle ∠ C = γ = 10.9354939616° = 10°56'6″ = 0.19108506998 rad

Height: ha = 9.31224212529
Height: hb = 8.87998491122
Height: hc = 45.53767983064

Median: ma = 26.68333281283
Median: mb = 22.77105950735
Median: mc = 47.52436783088

Inradius: r = 4.11655306492
Circumradius: R = 25.00655935101

Vertex coordinates: A[3; 3] B[12; 6] C[6; 52]
Centroid: CG[7; 20.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.88002420707; 4.11655306492]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.9388480468° = 111°56'19″ = 1.18878976097 rad
∠ B' = β' = 78.99664591483° = 78°59'47″ = 1.7632844344 rad
∠ C' = γ' = 169.0655060384° = 169°3'54″ = 0.19108506998 rad

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How did we calculate this triangle?

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

a = |BC| = |B-C| ; ; 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{ (12-6)**2 + (6-52)**2 } ; ; a = sqrt{ 2152 } = 46.39 ; ;

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

b = |AC| = |A-C| ; ; 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{ (3-6)**2 + (3-52)**2 } ; ; b = sqrt{ 2410 } = 49.09 ; ;

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

c = |AB| = |A-B| ; ; 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{ (3-12)**2 + (3-6)**2 } ; ; c = sqrt{ 90 } = 9.49 ; ;


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 = 46.39 ; ; b = 49.09 ; ; c = 9.49 ; ;

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

p = a+b+c = 46.39+49.09+9.49 = 104.97 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 104.97 }{ 2 } = 52.48 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 52.48 * (52.48-46.39)(52.48-49.09)(52.48-9.49) } ; ; T = sqrt{ 46656 } = 216 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 216 }{ 46.39 } = 9.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 216 }{ 49.09 } = 8.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 216 }{ 9.49 } = 45.54 ; ;

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{ 49.09**2+9.49**2-46.39**2 }{ 2 * 49.09 * 9.49 } ) = 68° 3'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46.39**2+9.49**2-49.09**2 }{ 2 * 46.39 * 9.49 } ) = 101° 13" ; ; gamma = 180° - alpha - beta = 180° - 68° 3'41" - 101° 13" = 10° 56'6" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 216 }{ 52.48 } = 4.12 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46.39 }{ 2 * sin 68° 3'41" } = 25.01 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 49.09**2+2 * 9.49**2 - 46.39**2 } }{ 2 } = 26.683 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.49**2+2 * 46.39**2 - 49.09**2 } }{ 2 } = 22.771 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 49.09**2+2 * 46.39**2 - 9.49**2 } }{ 2 } = 47.524 ; ;
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