Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 10.19880390272   b = 5.65768542495   c = 6.32545553203

Area: T = 16
Perimeter: p = 22.1799448597
Semiperimeter: s = 11.09897242985

Angle ∠ A = α = 116.5655051177° = 116°33'54″ = 2.03444439358 rad
Angle ∠ B = β = 29.74548812969° = 29°44'42″ = 0.51991461142 rad
Angle ∠ C = γ = 33.6990067526° = 33°41'24″ = 0.58880026035 rad

Height: ha = 3.13878581622
Height: hb = 5.65768542495
Height: hc = 5.06596442563

Median: ma = 3.16222776602
Median: mb = 8
Median: mc = 7.61657731059

Inradius: r = 1.44327770763
Circumradius: R = 5.70108771255

Vertex coordinates: A[-5; 4] B[1; 2] C[-9; 0]
Centroid: CG[-4.33333333333; 2]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.52548598835; 1.44327770763]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 63.43549488229° = 63°26'6″ = 2.03444439358 rad
∠ B' = β' = 150.2555118703° = 150°15'18″ = 0.51991461142 rad
∠ C' = γ' = 146.3109932474° = 146°18'36″ = 0.58880026035 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{ (1-(-9))**2 + (2-0)**2 } ; ; a = sqrt{ 104 } = 10.2 ; ;

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-(-9))**2 + (4-0)**2 } ; ; b = sqrt{ 32 } = 5.66 ; ;

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{ (-5-1)**2 + (4-2)**2 } ; ; c = sqrt{ 40 } = 6.32 ; ;


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 = 5.66 ; ; c = 6.32 ; ;

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

p = a+b+c = 10.2+5.66+6.32 = 22.18 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.18 }{ 2 } = 11.09 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.09 * (11.09-10.2)(11.09-5.66)(11.09-6.32) } ; ; T = sqrt{ 256 } = 16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16 }{ 10.2 } = 3.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16 }{ 5.66 } = 5.66 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16 }{ 6.32 } = 5.06 ; ;

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{ 10.2**2-5.66**2-6.32**2 }{ 2 * 5.66 * 6.32 } ) = 116° 33'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.66**2-10.2**2-6.32**2 }{ 2 * 10.2 * 6.32 } ) = 29° 44'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.32**2-10.2**2-5.66**2 }{ 2 * 5.66 * 10.2 } ) = 33° 41'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16 }{ 11.09 } = 1.44 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10.2 }{ 2 * sin 116° 33'54" } = 5.7 ; ;




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