Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 7.07110678119   b = 8.54440037453   c = 3.60655512755

Area: T = 12.5
Perimeter: p = 19.22106228326
Semiperimeter: s = 9.61103114163

Angle ∠ A = α = 54.24661127456° = 54°14'46″ = 0.94767732738 rad
Angle ∠ B = β = 101.3109932474° = 101°18'36″ = 1.76881918866 rad
Angle ∠ C = γ = 24.44439547804° = 24°26'38″ = 0.42766274931 rad

Height: ha = 3.53655339059
Height: hb = 2.92660286799
Height: hc = 6.93437524528

Median: ma = 5.52326805086
Median: mb = 3.64400549446
Median: mc = 7.63221687612

Inradius: r = 1.30106862586
Circumradius: R = 4.35766041822

Vertex coordinates: A[2; 7] B[4; 4] C[-1; -1]
Centroid: CG[1.66766666667; 3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.26601372517; 1.30106862586]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.7543887254° = 125°45'14″ = 0.94767732738 rad
∠ B' = β' = 78.6990067526° = 78°41'24″ = 1.76881918866 rad
∠ C' = γ' = 155.556604522° = 155°33'22″ = 0.42766274931 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{ (4-(-1))**2 + (4-(-1))**2 } ; ; a = sqrt{ 50 } = 7.07 ; ;

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{ (2-(-1))**2 + (7-(-1))**2 } ; ; b = sqrt{ 73 } = 8.54 ; ;

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{ (2-4)**2 + (7-4)**2 } ; ; c = sqrt{ 13 } = 3.61 ; ;
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 = 7.07 ; ; b = 8.54 ; ; c = 3.61 ; ;

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

p = a+b+c = 7.07+8.54+3.61 = 19.22 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.22 }{ 2 } = 9.61 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.61 * (9.61-7.07)(9.61-8.54)(9.61-3.61) } ; ; T = sqrt{ 156.25 } = 12.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.5 }{ 7.07 } = 3.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.5 }{ 8.54 } = 2.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.5 }{ 3.61 } = 6.93 ; ;

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{ 8.54**2+3.61**2-7.07**2 }{ 2 * 8.54 * 3.61 } ) = 54° 14'46" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.07**2+3.61**2-8.54**2 }{ 2 * 7.07 * 3.61 } ) = 101° 18'36" ; ; gamma = 180° - alpha - beta = 180° - 54° 14'46" - 101° 18'36" = 24° 26'38" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.5 }{ 9.61 } = 1.3 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.07 }{ 2 * sin 54° 14'46" } = 4.36 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 3.61**2 - 7.07**2 } }{ 2 } = 5.523 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.61**2+2 * 7.07**2 - 8.54**2 } }{ 2 } = 3.64 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 7.07**2 - 3.61**2 } }{ 2 } = 7.632 ; ;
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