Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 2.82884271247   b = 4.12331056256   c = 3.60655512755

Area: T = 5
Perimeter: p = 10.55770840258
Semiperimeter: s = 5.27985420129

Angle ∠ A = α = 42.27436890061° = 42°16'25″ = 0.73878150601 rad
Angle ∠ B = β = 78.6990067526° = 78°41'24″ = 1.37334007669 rad
Angle ∠ C = γ = 59.03662434679° = 59°2'10″ = 1.03303768265 rad

Height: ha = 3.53655339059
Height: hb = 2.42553562504
Height: hc = 2.77435009811

Median: ma = 3.60655512755
Median: mb = 2.5
Median: mc = 3.04113812651

Inradius: r = 0.94772312596
Circumradius: R = 2.10223796042

Vertex coordinates: A[2; -1] B[4; 2] C[6; 0]
Centroid: CG[4; 0.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.18994462519; 0.94772312596]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 137.7266310994° = 137°43'35″ = 0.73878150601 rad
∠ B' = β' = 101.3109932474° = 101°18'36″ = 1.37334007669 rad
∠ C' = γ' = 120.9643756532° = 120°57'50″ = 1.03303768265 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{ (4-6)**2 + (2-0)**2 } ; ; a = sqrt{ 8 } = 2.83 ; ;

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{ (2-6)**2 + (-1-0)**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{ (2-4)**2 + (-1-2)**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 = 2.83 ; ; b = 4.12 ; ; c = 3.61 ; ;

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

p = a+b+c = 2.83+4.12+3.61 = 10.56 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.56 }{ 2 } = 5.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.28 * (5.28-2.83)(5.28-4.12)(5.28-3.61) } ; ; T = sqrt{ 25 } = 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 * 5 }{ 2.83 } = 3.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5 }{ 4.12 } = 2.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5 }{ 3.61 } = 2.77 ; ;

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{ 2.83**2-4.12**2-3.61**2 }{ 2 * 4.12 * 3.61 } ) = 42° 16'25" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.12**2-2.83**2-3.61**2 }{ 2 * 2.83 * 3.61 } ) = 78° 41'24" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3.61**2-2.83**2-4.12**2 }{ 2 * 4.12 * 2.83 } ) = 59° 2'10" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5 }{ 5.28 } = 0.95 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.83 }{ 2 * sin 42° 16'25" } = 2.1 ; ;




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