Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.22195444573   b = 8.06222577483   c = 7.07110678119

Area: T = 27.5
Perimeter: p = 24.35328700175
Semiperimeter: s = 12.17664350087

Angle ∠ A = α = 74.74548812969° = 74°44'42″ = 1.30545442776 rad
Angle ∠ B = β = 57.52988077092° = 57°31'44″ = 1.00440671093 rad
Angle ∠ C = γ = 47.72663109939° = 47°43'35″ = 0.83329812667 rad

Height: ha = 5.966558759
Height: hb = 6.82219104024
Height: hc = 7.77881745931

Median: ma = 6.02107972894
Median: mb = 7.15989105316
Median: mc = 7.90656941504

Inradius: r = 2.25884607055
Circumradius: R = 4.7788135464

Vertex coordinates: A[-4; 5] B[1; 10] C[3; 1]
Centroid: CG[0; 5.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.43772022671; 2.25884607055]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.2555118703° = 105°15'18″ = 1.30545442776 rad
∠ B' = β' = 122.4711192291° = 122°28'16″ = 1.00440671093 rad
∠ C' = γ' = 132.2743689006° = 132°16'25″ = 0.83329812667 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{ (1-3)**2 + (10-1)**2 } ; ; a = sqrt{ 85 } = 9.22 ; ;

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{ (-4-3)**2 + (5-1)**2 } ; ; b = sqrt{ 65 } = 8.06 ; ;

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{ (-4-1)**2 + (5-10)**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 = 9.22 ; ; b = 8.06 ; ; c = 7.07 ; ;

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

p = a+b+c = 9.22+8.06+7.07 = 24.35 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.35 }{ 2 } = 12.18 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.18 * (12.18-9.22)(12.18-8.06)(12.18-7.07) } ; ; T = sqrt{ 756.25 } = 27.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27.5 }{ 9.22 } = 5.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27.5 }{ 8.06 } = 6.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27.5 }{ 7.07 } = 7.78 ; ;

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{ 8.06**2+7.07**2-9.22**2 }{ 2 * 8.06 * 7.07 } ) = 74° 44'42" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.22**2+7.07**2-8.06**2 }{ 2 * 9.22 * 7.07 } ) = 57° 31'44" ; ; gamma = 180° - alpha - beta = 180° - 74° 44'42" - 57° 31'44" = 47° 43'35" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27.5 }{ 12.18 } = 2.26 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.22 }{ 2 * sin 74° 44'42" } = 4.78 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.06**2+2 * 7.07**2 - 9.22**2 } }{ 2 } = 6.021 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.07**2+2 * 9.22**2 - 8.06**2 } }{ 2 } = 7.159 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.06**2+2 * 9.22**2 - 7.07**2 } }{ 2 } = 7.906 ; ;
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