Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 18.38547763109   b = 7.21111025509   c = 11.4021754251

Area: T = 13
Perimeter: p = 36.99876331128
Semiperimeter: s = 18.49988165564

Angle ∠ A = α = 161.5655051177° = 161°33'54″ = 2.82198420992 rad
Angle ∠ B = β = 7.12550163489° = 7°7'30″ = 0.12443549945 rad
Angle ∠ C = γ = 11.3109932474° = 11°18'36″ = 0.19773955598 rad

Height: ha = 1.41442135624
Height: hb = 3.60655512755
Height: hc = 2.28803508502

Median: ma = 2.55495097568
Median: mb = 14.86660687473
Median: mc = 12.7487548784

Inradius: r = 0.70327476574
Circumradius: R = 29.06988837075

Vertex coordinates: A[-2; -5] B[-9; 4] C[4; -9]
Centroid: CG[-2.33333333333; -3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[5.62219812593; 0.70327476574]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 18.43549488229° = 18°26'6″ = 2.82198420992 rad
∠ B' = β' = 172.8754983651° = 172°52'30″ = 0.12443549945 rad
∠ C' = γ' = 168.6990067526° = 168°41'24″ = 0.19773955598 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{ (-9-4)**2 + (4-(-9))**2 } ; ; a = sqrt{ 338 } = 18.38 ; ;

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{ (-2-4)**2 + (-5-(-9))**2 } ; ; b = sqrt{ 52 } = 7.21 ; ;

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{ (-2-(-9))**2 + (-5-4)**2 } ; ; c = sqrt{ 130 } = 11.4 ; ;


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 = 18.38 ; ; b = 7.21 ; ; c = 11.4 ; ;

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

p = a+b+c = 18.38+7.21+11.4 = 37 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-18.38)(18.5-7.21)(18.5-11.4) } ; ; T = sqrt{ 169 } = 13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13 }{ 18.38 } = 1.41 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13 }{ 7.21 } = 3.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13 }{ 11.4 } = 2.28 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 18.38**2-7.21**2-11.4**2 }{ 2 * 7.21 * 11.4 } ) = 161° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.21**2-18.38**2-11.4**2 }{ 2 * 18.38 * 11.4 } ) = 7° 7'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.4**2-18.38**2-7.21**2 }{ 2 * 7.21 * 18.38 } ) = 11° 18'36" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13 }{ 18.5 } = 0.7 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18.38 }{ 2 * sin 161° 33'54" } = 29.07 ; ;




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