Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 10.19880390272   b = 12.04215945788   c = 3.60655512755

Area: T = 17
Perimeter: p = 25.84551848814
Semiperimeter: s = 12.92325924407

Angle ∠ A = α = 51.54662907833° = 51°32'47″ = 0.98996524914 rad
Angle ∠ B = β = 112.3880135052° = 112°22'49″ = 1.96114033705 rad
Angle ∠ C = γ = 16.07435741647° = 16°4'25″ = 0.28105367917 rad

Height: ha = 3.33439742973
Height: hb = 2.8243546315
Height: hc = 9.43299033358

Median: ma = 7.28801098893
Median: mb = 4.7176990566
Median: mc = 11.01113577728

Inradius: r = 1.31655255091
Circumradius: R = 6.51112359634

Vertex coordinates: A[-9; 2] B[-7; -1] C[3; 1]
Centroid: CG[-4.33333333333; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0.54216869744; 1.31655255091]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128.4543709217° = 128°27'13″ = 0.98996524914 rad
∠ B' = β' = 67.6219864948° = 67°37'11″ = 1.96114033705 rad
∠ C' = γ' = 163.9266425835° = 163°55'35″ = 0.28105367917 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{ (-7-3)**2 + (-1-1)**2 } ; ; a = sqrt{ 104 } = 10.2 ; ;

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{ (-9-3)**2 + (2-1)**2 } ; ; b = sqrt{ 145 } = 12.04 ; ;

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-(-7))**2 + (2-(-1))**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 = 10.2 ; ; b = 12.04 ; ; c = 3.61 ; ;

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

p = a+b+c = 10.2+12.04+3.61 = 25.85 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.85 }{ 2 } = 12.92 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.92 * (12.92-10.2)(12.92-12.04)(12.92-3.61) } ; ; T = sqrt{ 289 } = 17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17 }{ 10.2 } = 3.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17 }{ 12.04 } = 2.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17 }{ 3.61 } = 9.43 ; ;

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{ 12.04**2+3.61**2-10.2**2 }{ 2 * 12.04 * 3.61 } ) = 51° 32'47" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.2**2+3.61**2-12.04**2 }{ 2 * 10.2 * 3.61 } ) = 112° 22'49" ; ; gamma = 180° - alpha - beta = 180° - 51° 32'47" - 112° 22'49" = 16° 4'25" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17 }{ 12.92 } = 1.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.2 }{ 2 * sin 51° 32'47" } = 6.51 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.04**2+2 * 3.61**2 - 10.2**2 } }{ 2 } = 7.28 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.61**2+2 * 10.2**2 - 12.04**2 } }{ 2 } = 4.717 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.04**2+2 * 10.2**2 - 3.61**2 } }{ 2 } = 11.011 ; ;
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