Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 27.73108492477   b = 25.4955097568   c = 10.63301458127

Area: T = 135.5
Perimeter: p = 63.85660926284
Semiperimeter: s = 31.92880463142

Angle ∠ A = α = 90.63442447144° = 90°38'3″ = 1.58218659853 rad
Angle ∠ B = β = 66.827693099° = 66°49'37″ = 1.16663499748 rad
Angle ∠ C = γ = 22.53988242956° = 22°32'20″ = 0.39333766935 rad

Height: ha = 9.7732509943
Height: hb = 10.62994945245
Height: hc = 25.49435355332

Median: ma = 13.75768164922
Median: mb = 16.68883192683
Median: mc = 26.10107662723

Inradius: r = 4.24439176725
Circumradius: R = 13.86662741834

Vertex coordinates: A[-5; 2] B[2; 10] C[14; -15]
Centroid: CG[3.66766666667; -1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.81765846864; 4.24439176725]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 89.36657552856° = 89°21'57″ = 1.58218659853 rad
∠ B' = β' = 113.173306901° = 113°10'23″ = 1.16663499748 rad
∠ C' = γ' = 157.4611175704° = 157°27'40″ = 0.39333766935 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{ (2-14)**2 + (10-(-15))**2 } ; ; a = sqrt{ 769 } = 27.73 ; ;

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{ (-5-14)**2 + (2-(-15))**2 } ; ; b = sqrt{ 650 } = 25.5 ; ;

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{ (-5-2)**2 + (2-10)**2 } ; ; c = sqrt{ 113 } = 10.63 ; ;


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 = 27.73 ; ; b = 25.5 ; ; c = 10.63 ; ;

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

p = a+b+c = 27.73+25.5+10.63 = 63.86 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 63.86 }{ 2 } = 31.93 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31.93 * (31.93-27.73)(31.93-25.5)(31.93-10.63) } ; ; T = sqrt{ 18360.25 } = 135.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 135.5 }{ 27.73 } = 9.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 135.5 }{ 25.5 } = 10.63 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 135.5 }{ 10.63 } = 25.49 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 27.73**2-25.5**2-10.63**2 }{ 2 * 25.5 * 10.63 } ) = 90° 38'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 25.5**2-27.73**2-10.63**2 }{ 2 * 27.73 * 10.63 } ) = 66° 49'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.63**2-27.73**2-25.5**2 }{ 2 * 25.5 * 27.73 } ) = 22° 32'20" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 135.5 }{ 31.93 } = 4.24 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27.73 }{ 2 * sin 90° 38'3" } = 13.87 ; ;




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