Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.944427191   b = 13.34216640641   c = 5.09990195136

Area: T = 14
Perimeter: p = 27.38549554877
Semiperimeter: s = 13.69224777439

Angle ∠ A = α = 24.30545492659° = 24°18'16″ = 0.42441944079 rad
Angle ∠ B = β = 142.1255016349° = 142°7'30″ = 2.48105494847 rad
Angle ∠ C = γ = 13.57704343852° = 13°34'14″ = 0.23768487609 rad

Height: ha = 3.13304951685
Height: hb = 2.09986887292
Height: hc = 5.49112517839

Median: ma = 9.05553851381
Median: mb = 2.91554759474
Median: mc = 11.06879718106

Inradius: r = 1.02224592117
Circumradius: R = 10.86655982113

Vertex coordinates: A[-9; -6] B[-4; -5] C[4; -9]
Centroid: CG[-3; -6.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-1.3154590415; 1.02224592117]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.6955450734° = 155°41'44″ = 0.42441944079 rad
∠ B' = β' = 37.87549836511° = 37°52'30″ = 2.48105494847 rad
∠ C' = γ' = 166.4329565615° = 166°25'46″ = 0.23768487609 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{ (-4-4)**2 + (-5-(-9))**2 } ; ; a = sqrt{ 80 } = 8.94 ; ;

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{ (-9-4)**2 + (-6-(-9))**2 } ; ; b = sqrt{ 178 } = 13.34 ; ;

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{ (-9-(-4))**2 + (-6-(-5))**2 } ; ; c = sqrt{ 26 } = 5.1 ; ;


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 = 8.94 ; ; b = 13.34 ; ; c = 5.1 ; ;

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

p = a+b+c = 8.94+13.34+5.1 = 27.38 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.38 }{ 2 } = 13.69 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.69 * (13.69-8.94)(13.69-13.34)(13.69-5.1) } ; ; T = sqrt{ 196 } = 14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14 }{ 8.94 } = 3.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14 }{ 13.34 } = 2.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14 }{ 5.1 } = 5.49 ; ;

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{ 13.34**2+5.1**2-8.94**2 }{ 2 * 13.34 * 5.1 } ) = 24° 18'16" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.94**2+5.1**2-13.34**2 }{ 2 * 8.94 * 5.1 } ) = 142° 7'30" ; ; gamma = 180° - alpha - beta = 180° - 24° 18'16" - 142° 7'30" = 13° 34'14" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14 }{ 13.69 } = 1.02 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.94 }{ 2 * sin 24° 18'16" } = 10.87 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.34**2+2 * 5.1**2 - 8.94**2 } }{ 2 } = 9.055 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 8.94**2 - 13.34**2 } }{ 2 } = 2.915 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.34**2+2 * 8.94**2 - 5.1**2 } }{ 2 } = 11.068 ; ;
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