Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 11.04553610172   b = 5.65768542495   c = 8.6022325267

Area: T = 24
Perimeter: p = 25.30545405337
Semiperimeter: s = 12.65222702669

Angle ∠ A = α = 99.4622322208° = 99°27'44″ = 1.73659450042 rad
Angle ∠ B = β = 30.34332488842° = 30°20'36″ = 0.53295895988 rad
Angle ∠ C = γ = 50.19444289077° = 50°11'40″ = 0.87660580506 rad

Height: ha = 4.346571581
Height: hb = 8.48552813742
Height: hc = 5.58798866597

Median: ma = 4.74334164903
Median: mb = 9.48768329805
Median: mc = 7.64985292704

Inradius: r = 1.89768927705
Circumradius: R = 5.59988590107

Vertex coordinates: A[1; -1] B[-4; 6] C[-3; -5]
Centroid: CG[-2; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.24105251497; 1.89768927705]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 80.5387677792° = 80°32'16″ = 1.73659450042 rad
∠ B' = β' = 149.6576751116° = 149°39'24″ = 0.53295895988 rad
∠ C' = γ' = 129.8065571092° = 129°48'20″ = 0.87660580506 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-(-3))**2 + (6-(-5))**2 } ; ; a = sqrt{ 122 } = 11.05 ; ;

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{ (1-(-3))**2 + (-1-(-5))**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{ (1-(-4))**2 + (-1-6)**2 } ; ; c = sqrt{ 74 } = 8.6 ; ;


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 = 11.05 ; ; b = 5.66 ; ; c = 8.6 ; ;

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

p = a+b+c = 11.05+5.66+8.6 = 25.3 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.3 }{ 2 } = 12.65 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.65 * (12.65-11.05)(12.65-5.66)(12.65-8.6) } ; ; T = sqrt{ 576 } = 24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24 }{ 11.05 } = 4.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24 }{ 5.66 } = 8.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24 }{ 8.6 } = 5.58 ; ;

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+8.6**2-11.05**2 }{ 2 * 5.66 * 8.6 } ) = 99° 27'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 11.05**2+8.6**2-5.66**2 }{ 2 * 11.05 * 8.6 } ) = 30° 20'36" ; ; gamma = 180° - alpha - beta = 180° - 99° 27'44" - 30° 20'36" = 50° 11'40" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24 }{ 12.65 } = 1.9 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 11.05 }{ 2 * sin 99° 27'44" } = 5.6 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.66**2+2 * 8.6**2 - 11.05**2 } }{ 2 } = 4.743 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.6**2+2 * 11.05**2 - 5.66**2 } }{ 2 } = 9.487 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.66**2+2 * 11.05**2 - 8.6**2 } }{ 2 } = 7.649 ; ;
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