Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 9.48768329805   b = 6.08327625303   c = 3.60655512755

Area: T = 4.5
Perimeter: p = 19.17551467863
Semiperimeter: s = 9.58875733931

Angle ∠ A = α = 155.7722254682° = 155°46'20″ = 2.71987387275 rad
Angle ∠ B = β = 15.25551187031° = 15°15'18″ = 0.26662520492 rad
Angle ∠ C = γ = 8.97326266149° = 8°58'21″ = 0.1576601877 rad

Height: ha = 0.94986832981
Height: hb = 1.48795908857
Height: hc = 2.4966150883

Median: ma = 1.58111388301
Median: mb = 6.5
Median: mc = 7.76220873481

Inradius: r = 0.46993575544
Circumradius: R = 11.55990272563

Vertex coordinates: A[0; -3] B[-2; 0] C[1; -9]
Centroid: CG[-0.33333333333; -4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.72109776993; 0.46993575544]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 24.2287745318° = 24°13'40″ = 2.71987387275 rad
∠ B' = β' = 164.7454881297° = 164°44'42″ = 0.26662520492 rad
∠ C' = γ' = 171.0277373385° = 171°1'39″ = 0.1576601877 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{ (-2-1)**2 + (0-(-9))**2 } ; ; a = sqrt{ 90 } = 9.49 ; ;

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{ (0-1)**2 + (-3-(-9))**2 } ; ; b = sqrt{ 37 } = 6.08 ; ;

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{ (0-(-2))**2 + (-3-0)**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 = 9.49 ; ; b = 6.08 ; ; c = 3.61 ; ;

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

p = a+b+c = 9.49+6.08+3.61 = 19.18 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.18 }{ 2 } = 9.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.59 * (9.59-9.49)(9.59-6.08)(9.59-3.61) } ; ; T = sqrt{ 20.25 } = 4.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4.5 }{ 9.49 } = 0.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.5 }{ 6.08 } = 1.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.5 }{ 3.61 } = 2.5 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 9.49**2-6.08**2-3.61**2 }{ 2 * 6.08 * 3.61 } ) = 155° 46'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6.08**2-9.49**2-3.61**2 }{ 2 * 9.49 * 3.61 } ) = 15° 15'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.61**2-9.49**2-6.08**2 }{ 2 * 6.08 * 9.49 } ) = 8° 58'21" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.5 }{ 9.59 } = 0.47 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.49 }{ 2 * sin 155° 46'20" } = 11.56 ; ;




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