Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 9.43439811321   b = 32.38882694814   c = 40

Area: T = 100
Perimeter: p = 81.82222506135
Semiperimeter: s = 40.91111253067

Angle ∠ A = α = 8.88106591505° = 8°52'50″ = 0.15549967419 rad
Angle ∠ B = β = 32.00553832081° = 32°19″ = 0.55985993153 rad
Angle ∠ C = γ = 139.1143957641° = 139°6'50″ = 2.42879965963 rad

Height: ha = 21.21999576001
Height: hb = 6.17550752109
Height: hc = 5

Median: ma = 36.08767011515
Median: mb = 24.1329857024
Median: mc = 13

Inradius: r = 2.4444322889
Circumradius: R = 30.55550323188

Vertex coordinates: A[-21; 0] B[19; 0] C[11; -5]
Centroid: CG[3; -1.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[3.91109166223; 2.4444322889]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.1199340849° = 171°7'10″ = 0.15549967419 rad
∠ B' = β' = 147.9954616792° = 147°59'41″ = 0.55985993153 rad
∠ C' = γ' = 40.88660423586° = 40°53'10″ = 2.42879965963 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (19-11)**2 + (0-(-5))**2 } ; ; a = sqrt{ 89 } = 9.43 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (-21-11)**2 + (0-(-5))**2 } ; ; b = sqrt{ 1049 } = 32.39 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (-21-19)**2 + (0-0)**2 } ; ; c = sqrt{ 1600 } = 40 ; ;


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 = 9.43 ; ; b = 32.39 ; ; c = 40 ; ;

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

p = a+b+c = 9.43+32.39+40 = 81.82 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 81.82 }{ 2 } = 40.91 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.91 * (40.91-9.43)(40.91-32.39)(40.91-40) } ; ; T = sqrt{ 10000 } = 100 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 100 }{ 9.43 } = 21.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 100 }{ 32.39 } = 6.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 100 }{ 40 } = 5 ; ;

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{ 32.39**2+40**2-9.43**2 }{ 2 * 32.39 * 40 } ) = 8° 52'50" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.43**2+40**2-32.39**2 }{ 2 * 9.43 * 40 } ) = 32° 19" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 9.43**2+32.39**2-40**2 }{ 2 * 9.43 * 32.39 } ) = 139° 6'50" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 100 }{ 40.91 } = 2.44 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.43 }{ 2 * sin 8° 52'50" } = 30.56 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.39**2+2 * 40**2 - 9.43**2 } }{ 2 } = 36.087 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 9.43**2 - 32.39**2 } }{ 2 } = 24.13 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 32.39**2+2 * 9.43**2 - 40**2 } }{ 2 } = 13 ; ;
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