Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 4.24326406871   b = 4.12331056256   c = 4.12331056256

Area: T = 7.5
Perimeter: p = 12.48988519384
Semiperimeter: s = 6.24444259692

Angle ∠ A = α = 61.92875130641° = 61°55'39″ = 1.08108390005 rad
Angle ∠ B = β = 59.03662434679° = 59°2'10″ = 1.03303768265 rad
Angle ∠ C = γ = 59.03662434679° = 59°2'10″ = 1.03303768265 rad

Height: ha = 3.53655339059
Height: hb = 3.63880343755
Height: hc = 3.63880343755

Median: ma = 3.53655339059
Median: mb = 3.64400549446
Median: mc = 3.64400549446

Inradius: r = 1.20110711692
Circumradius: R = 2.4044163056

Vertex coordinates: A[3; 2] B[4; 6] C[7; 3]
Centroid: CG[4.66766666667; 3.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.72106427015; 1.20110711692]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 118.0722486936° = 118°4'21″ = 1.08108390005 rad
∠ B' = β' = 120.9643756532° = 120°57'50″ = 1.03303768265 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 = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 } ; ; a = sqrt{ (4-7)**2 + (6-3)**2 } ; ; a = sqrt{ 18 } = 4.24 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 } ; ; b = sqrt{ (3-7)**2 + (2-3)**2 } ; ; b = sqrt{ 17 } = 4.12 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 } ; ; c = sqrt{ (3-4)**2 + (2-6)**2 } ; ; c = sqrt{ 17 } = 4.12 ; ;


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

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

p = a+b+c = 4.24+4.12+4.12 = 12.49 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.49 }{ 2 } = 6.24 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.24 * (6.24-4.24)(6.24-4.12)(6.24-4.12) } ; ; T = sqrt{ 56.25 } = 7.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.5 }{ 4.24 } = 3.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.5 }{ 4.12 } = 3.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.5 }{ 4.12 } = 3.64 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 4.12**2+4.12**2-4.24**2 }{ 2 * 4.12 * 4.12 } ) = 61° 55'39" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.24**2+4.12**2-4.12**2 }{ 2 * 4.24 * 4.12 } ) = 59° 2'10" ; ; gamma = 180° - alpha - beta = 180° - 61° 55'39" - 59° 2'10" = 59° 2'10" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.5 }{ 6.24 } = 1.2 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.24 }{ 2 * sin 61° 55'39" } = 2.4 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 4.12**2 - 4.24**2 } }{ 2 } = 3.536 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 4.24**2 - 4.12**2 } }{ 2 } = 3.64 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.12**2+2 * 4.24**2 - 4.12**2 } }{ 2 } = 3.64 ; ;
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