Triangle calculator VC

Please enter the coordinates of the three vertices


Right scalene triangle.

Sides: a = 18.11107702763   b = 20.24884567313   c = 9.05553851381

Area: T = 82
Perimeter: p = 47.41546121457
Semiperimeter: s = 23.70773060729

Angle ∠ A = α = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 26.56550511771° = 26°33'54″ = 0.4643647609 rad

Height: ha = 9.05553851381
Height: hb = 8.09993826925
Height: hc = 18.11107702763

Median: ma = 12.80662484749
Median: mb = 10.12442283657
Median: mc = 18.66881547026

Inradius: r = 3.45988493415
Circumradius: R = 10.12442283657

Vertex coordinates: A[-2; 3] B[7; 4] C[5; 22]
Centroid: CG[3.33333333333; 9.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0; 3.45988493415]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 153.4354948823° = 153°26'6″ = 0.4643647609 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{ (7-5)**2 + (4-22)**2 } ; ; a = sqrt{ 328 } = 18.11 ; ;

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{ (-2-5)**2 + (3-22)**2 } ; ; b = sqrt{ 410 } = 20.25 ; ;

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{ (-2-7)**2 + (3-4)**2 } ; ; c = sqrt{ 82 } = 9.06 ; ;


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 = 18.11 ; ; b = 20.25 ; ; c = 9.06 ; ;

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

p = a+b+c = 18.11+20.25+9.06 = 47.41 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47.41 }{ 2 } = 23.71 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.71 * (23.71-18.11)(23.71-20.25)(23.71-9.06) } ; ; T = sqrt{ 6724 } = 82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 82 }{ 18.11 } = 9.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 82 }{ 20.25 } = 8.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 82 }{ 9.06 } = 18.11 ; ;

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{ 20.25**2+9.06**2-18.11**2 }{ 2 * 20.25 * 9.06 } ) = 63° 26'6" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.11**2+9.06**2-20.25**2 }{ 2 * 18.11 * 9.06 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 63° 26'6" - 90° = 26° 33'54" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 82 }{ 23.71 } = 3.46 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18.11 }{ 2 * sin 63° 26'6" } = 10.12 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.25**2+2 * 9.06**2 - 18.11**2 } }{ 2 } = 12.806 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.06**2+2 * 18.11**2 - 20.25**2 } }{ 2 } = 10.124 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 20.25**2+2 * 18.11**2 - 9.06**2 } }{ 2 } = 18.668 ; ;
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