Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene Pythagorean triangle.

Sides: a = 4.24326406871   b = 5.65768542495   c = 7.07110678119

Area: T = 12
Perimeter: p = 16.97105627485
Semiperimeter: s = 8.48552813742

Angle ∠ A = α = 36.87698976458° = 36°52'12″ = 0.64435011088 rad
Angle ∠ B = β = 53.13301023542° = 53°7'48″ = 0.9277295218 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 5.65768542495
Height: hb = 4.24326406871
Height: hc = 3.39441125497

Median: ma = 6.04215229868
Median: mb = 5.09990195136
Median: mc = 3.53655339059

Inradius: r = 1.41442135624
Circumradius: R = 3.53655339059

Vertex coordinates: A[0; 0] B[7; 1] C[4; 4]
Centroid: CG[3.66766666667; 1.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.06106601718; 1.41442135624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.1330102354° = 143°7'48″ = 0.64435011088 rad
∠ B' = β' = 126.8769897646° = 126°52'12″ = 0.9277295218 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{ (7-4)**2 + (1-4)**2 } ; ; a = sqrt{ 18 } = 4.24 ; ;

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{ (0-4)**2 + (0-4)**2 } ; ; b = sqrt{ 32 } = 5.66 ; ;

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{ (0-7)**2 + (0-1)**2 } ; ; c = sqrt{ 50 } = 7.07 ; ;


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 = 4.24 ; ; b = 5.66 ; ; c = 7.07 ; ;

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

p = a+b+c = 4.24+5.66+7.07 = 16.97 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.97 }{ 2 } = 8.49 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.49 * (8.49-4.24)(8.49-5.66)(8.49-7.07) } ; ; T = sqrt{ 144 } = 12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12 }{ 4.24 } = 5.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12 }{ 5.66 } = 4.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12 }{ 7.07 } = 3.39 ; ;

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{ 5.66**2+7.07**2-4.24**2 }{ 2 * 5.66 * 7.07 } ) = 36° 52'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.24**2+7.07**2-5.66**2 }{ 2 * 4.24 * 7.07 } ) = 53° 7'48" ; ; gamma = 180° - alpha - beta = 180° - 36° 52'12" - 53° 7'48" = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12 }{ 8.49 } = 1.41 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.24 }{ 2 * sin 36° 52'12" } = 3.54 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.66**2+2 * 7.07**2 - 4.24**2 } }{ 2 } = 6.042 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.07**2+2 * 4.24**2 - 5.66**2 } }{ 2 } = 5.099 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.66**2+2 * 4.24**2 - 7.07**2 } }{ 2 } = 3.536 ; ;
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