Triangle calculator VC

Please enter the coordinates of the three vertices


Acute isosceles triangle.

Sides: a = 5.09990195136   b = 5.09990195136   c = 3.74216573868

Area: T = 8.87441196746
Perimeter: p = 13.9439696414
Semiperimeter: s = 6.9769848207

Angle ∠ A = α = 68.47554600999° = 68°28'32″ = 1.19551222356 rad
Angle ∠ B = β = 68.47554600999° = 68°28'32″ = 1.19551222356 rad
Angle ∠ C = γ = 43.04990798002° = 43°2'57″ = 0.75113481825 rad

Height: ha = 3.48107161067
Height: hb = 3.48107161067
Height: hc = 4.74334164903

Median: ma = 3.67442346142
Median: mb = 3.67442346142
Median: mc = 4.74334164903

Inradius: r = 1.27332156298
Circumradius: R = 2.74106406388

Vertex coordinates: A[3; 4; 1] B[0; 6; 2] C[3; 5; 6]
Centroid: CG[2; 5; 3]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 111.52545399° = 111°31'28″ = 1.19551222356 rad
∠ B' = β' = 111.52545399° = 111°31'28″ = 1.19551222356 rad
∠ C' = γ' = 136.95109202° = 136°57'3″ = 0.75113481825 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 + (B_z-C_z)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 + (B_z-C_z)**2 } ; ; a = sqrt{ (0-3)**2 + (6-5)**2 + (2 - 6)**2 } ; ; a = sqrt{ 26 } = 5.1 ; ;

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 + (A_z-C_z)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 + (A_z-C_z)**2 } ; ; b = sqrt{ (3-3)**2 + (4-5)**2 + (1 - 6)**2 } ; ; b = sqrt{ 26 } = 5.1 ; ;

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 + (A_z-B_z)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 + (A_z-B_z)**2 } ; ; c = sqrt{ (3-0)**2 + (4-6)**2 + (1 - 2)**2 } ; ; c = sqrt{ 14 } = 3.74 ; ;


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 = 5.1 ; ; b = 5.1 ; ; c = 3.74 ; ;

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

p = a+b+c = 5.1+5.1+3.74 = 13.94 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.94 }{ 2 } = 6.97 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.97 * (6.97-5.1)(6.97-5.1)(6.97-3.74) } ; ; T = sqrt{ 78.75 } = 8.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.87 }{ 5.1 } = 3.48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.87 }{ 5.1 } = 3.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.87 }{ 3.74 } = 4.74 ; ;

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{ 5.1**2+3.74**2-5.1**2 }{ 2 * 5.1 * 3.74 } ) = 68° 28'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.1**2+3.74**2-5.1**2 }{ 2 * 5.1 * 3.74 } ) = 68° 28'32" ; ; gamma = 180° - alpha - beta = 180° - 68° 28'32" - 68° 28'32" = 43° 2'57" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.87 }{ 6.97 } = 1.27 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.1 }{ 2 * sin 68° 28'32" } = 2.74 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 3.74**2 - 5.1**2 } }{ 2 } = 3.674 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.74**2+2 * 5.1**2 - 5.1**2 } }{ 2 } = 3.674 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.1**2+2 * 5.1**2 - 3.74**2 } }{ 2 } = 4.743 ; ;
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