Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 8.944427191   b = 2.23660679775   c = 9.22195444573

Area: T = 10
Perimeter: p = 20.43998843448
Semiperimeter: s = 10.21999421724

Angle ∠ A = α = 75.96437565321° = 75°57'50″ = 1.32658176637 rad
Angle ∠ B = β = 14.03662434679° = 14°2'10″ = 0.24549786631 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 2.23660679775
Height: hb = 8.944427191
Height: hc = 2.16993045782

Median: ma = 5
Median: mb = 9.01438781887
Median: mc = 4.61097722286

Inradius: r = 0.98803977151
Circumradius: R = 4.61097722286

Vertex coordinates: A[1; 4] B[10; 6] C[2; 2]
Centroid: CG[4.33333333333; 4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.92215908604; 0.98803977151]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 104.0366243468° = 104°2'10″ = 1.32658176637 rad
∠ B' = β' = 165.9643756532° = 165°57'50″ = 0.24549786631 rad
∠ C' = γ' = 90° = 1.57107963268 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{ (10-2)**2 + (6-2)**2 } ; ; a = sqrt{ 80 } = 8.94 ; ;

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{ (1-2)**2 + (4-2)**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (1-10)**2 + (4-6)**2 } ; ; c = sqrt{ 85 } = 9.22 ; ;
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 = 8.94 ; ; b = 2.24 ; ; c = 9.22 ; ;

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

p = a+b+c = 8.94+2.24+9.22 = 20.4 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.4 }{ 2 } = 10.2 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.2 * (10.2-8.94)(10.2-2.24)(10.2-9.22) } ; ; T = sqrt{ 100 } = 10 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10 }{ 8.94 } = 2.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10 }{ 2.24 } = 8.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10 }{ 9.22 } = 2.17 ; ;

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{ 2.24**2+9.22**2-8.94**2 }{ 2 * 2.24 * 9.22 } ) = 75° 57'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.94**2+9.22**2-2.24**2 }{ 2 * 8.94 * 9.22 } ) = 14° 2'10" ; ; gamma = 180° - alpha - beta = 180° - 75° 57'50" - 14° 2'10" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10 }{ 10.2 } = 0.98 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.94 }{ 2 * sin 75° 57'50" } = 4.61 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 9.22**2 - 8.94**2 } }{ 2 } = 5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.22**2+2 * 8.94**2 - 2.24**2 } }{ 2 } = 9.014 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 8.94**2 - 9.22**2 } }{ 2 } = 4.61 ; ;
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