Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 6.40331242374   b = 8.48552813742   c = 10.05498756211

Area: T = 27
Perimeter: p = 24.93882812328
Semiperimeter: s = 12.46991406164

Angle ∠ A = α = 39.28994068625° = 39°17'22″ = 0.68657295109 rad
Angle ∠ B = β = 57.05107848834° = 57°3'3″ = 0.99657240371 rad
Angle ∠ C = γ = 83.66598082541° = 83°39'35″ = 1.46601391056 rad

Height: ha = 8.4333383142
Height: hb = 6.36439610307
Height: hc = 5.37332008271

Median: ma = 8.73221245983
Median: mb = 7.28801098893
Median: mc = 5.59901699437

Inradius: r = 2.16553456987
Circumradius: R = 5.05658607966

Vertex coordinates: A[1; 3] B[11; 4] C[7; 9]
Centroid: CG[6.33333333333; 5.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.40334648047; 2.16553456987]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.7110593137° = 140°42'38″ = 0.68657295109 rad
∠ B' = β' = 122.9499215117° = 122°56'57″ = 0.99657240371 rad
∠ C' = γ' = 96.34401917459° = 96°20'25″ = 1.46601391056 rad

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How did we calculate this triangle?

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

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (11-7)**2 + (4-9)**2 } ; ; a = sqrt{ 41 } = 6.4 ; ;

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

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (1-7)**2 + (3-9)**2 } ; ; b = sqrt{ 72 } = 8.49 ; ;

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

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (1-11)**2 + (3-4)**2 } ; ; c = sqrt{ 101 } = 10.05 ; ;


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 = 6.4 ; ; b = 8.49 ; ; c = 10.05 ; ;

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

p = a+b+c = 6.4+8.49+10.05 = 24.94 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.94 }{ 2 } = 12.47 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.47 * (12.47-6.4)(12.47-8.49)(12.47-10.05) } ; ; T = sqrt{ 729 } = 27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27 }{ 6.4 } = 8.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27 }{ 8.49 } = 6.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27 }{ 10.05 } = 5.37 ; ;

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{ 8.49**2+10.05**2-6.4**2 }{ 2 * 8.49 * 10.05 } ) = 39° 17'22" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.4**2+10.05**2-8.49**2 }{ 2 * 6.4 * 10.05 } ) = 57° 3'3" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 6.4**2+8.49**2-10.05**2 }{ 2 * 6.4 * 8.49 } ) = 83° 39'35" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27 }{ 12.47 } = 2.17 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.4 }{ 2 * sin 39° 17'22" } = 5.06 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.49**2+2 * 10.05**2 - 6.4**2 } }{ 2 } = 8.732 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.05**2+2 * 6.4**2 - 8.49**2 } }{ 2 } = 7.28 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.49**2+2 * 6.4**2 - 10.05**2 } }{ 2 } = 5.59 ; ;
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