Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 8.06222577483   b = 4.12331056256   c = 4.24326406871

Area: T = 4.5
Perimeter: p = 16.4288004061
Semiperimeter: s = 8.21440020305

Angle ∠ A = α = 149.0366243468° = 149°2'10″ = 2.60111731533 rad
Angle ∠ B = β = 15.25551187031° = 15°15'18″ = 0.26662520492 rad
Angle ∠ C = γ = 15.7098637829° = 15°42'31″ = 0.27441674511 rad

Height: ha = 1.11663126113
Height: hb = 2.18328206253
Height: hc = 2.12113203436

Median: ma = 1.11880339887
Median: mb = 6.10332778079
Median: mc = 6.04215229868

Inradius: r = 0.5487845007
Circumradius: R = 7.83551061824

Vertex coordinates: A[4; 1] B[1; -2] C[5; 5]
Centroid: CG[3.33333333333; 1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.00987650257; 0.5487845007]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 30.96437565321° = 30°57'50″ = 2.60111731533 rad
∠ B' = β' = 164.7454881297° = 164°44'42″ = 0.26662520492 rad
∠ C' = γ' = 164.2911362171° = 164°17'29″ = 0.27441674511 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{ (1-5)**2 + (-2-5)**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (4-5)**2 + (1-5)**2 } ; ; b = sqrt{ 17 } = 4.12 ; ;

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{ (4-1)**2 + (1-(-2))**2 } ; ; c = sqrt{ 18 } = 4.24 ; ;


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 = 8.06 ; ; b = 4.12 ; ; c = 4.24 ; ;

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

p = a+b+c = 8.06+4.12+4.24 = 16.43 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.43 }{ 2 } = 8.21 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.21 * (8.21-8.06)(8.21-4.12)(8.21-4.24) } ; ; 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 }{ 8.06 } = 1.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4.5 }{ 4.12 } = 2.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4.5 }{ 4.24 } = 2.12 ; ;

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{ 8.06**2-4.12**2-4.24**2 }{ 2 * 4.12 * 4.24 } ) = 149° 2'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.12**2-8.06**2-4.24**2 }{ 2 * 8.06 * 4.24 } ) = 15° 15'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.24**2-8.06**2-4.12**2 }{ 2 * 4.12 * 8.06 } ) = 15° 42'31" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4.5 }{ 8.21 } = 0.55 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.06 }{ 2 * sin 149° 2'10" } = 7.84 ; ;




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