Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.11104335791   b = 8.6022325267   c = 4.5832575695

Area: T = 19.48107597388
Perimeter: p = 22.29553345411
Semiperimeter: s = 11.14876672706

Angle ∠ A = α = 81.2455333311° = 81°14'43″ = 1.41879985682 rad
Angle ∠ B = β = 68.94435313444° = 68°56'37″ = 1.20332916199 rad
Angle ∠ C = γ = 29.81111353446° = 29°48'40″ = 0.52203024655 rad

Height: ha = 4.27765823535
Height: hb = 4.52991846411
Height: hc = 8.50221005808

Median: ma = 5.17220402164
Median: mb = 5.78879184514
Median: mc = 8.55986213843

Inradius: r = 1.74875189442
Circumradius: R = 4.60989147162

Vertex coordinates: A[2; 3; 1] B[4; 7; 2] C[-5; 8; 1]
Centroid: CG[0.33333333333; 6; 1.33333333333]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 98.7554666689° = 98°45'17″ = 1.41879985682 rad
∠ B' = β' = 111.0566468656° = 111°3'23″ = 1.20332916199 rad
∠ C' = γ' = 150.1898864655° = 150°11'20″ = 0.52203024655 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{ (4-(-5))**2 + (7-8)**2 + (2 - 1)**2 } ; ; a = sqrt{ 83 } = 9.11 ; ;

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{ (2-(-5))**2 + (3-8)**2 + (1 - 1)**2 } ; ; b = sqrt{ 74 } = 8.6 ; ;

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{ (2-4)**2 + (3-7)**2 + (1 - 2)**2 } ; ; c = sqrt{ 21 } = 4.58 ; ;
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 = 9.11 ; ; b = 8.6 ; ; c = 4.58 ; ;

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

p = a+b+c = 9.11+8.6+4.58 = 22.3 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.3 }{ 2 } = 11.15 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.15 * (11.15-9.11)(11.15-8.6)(11.15-4.58) } ; ; T = sqrt{ 379.5 } = 19.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19.48 }{ 9.11 } = 4.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19.48 }{ 8.6 } = 4.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19.48 }{ 4.58 } = 8.5 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 8.6**2+4.58**2-9.11**2 }{ 2 * 8.6 * 4.58 } ) = 81° 14'43" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.11**2+4.58**2-8.6**2 }{ 2 * 9.11 * 4.58 } ) = 68° 56'37" ; ;
 gamma = 180° - alpha - beta = 180° - 81° 14'43" - 68° 56'37" = 29° 48'40" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19.48 }{ 11.15 } = 1.75 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.11 }{ 2 * sin 81° 14'43" } = 4.61 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.6**2+2 * 4.58**2 - 9.11**2 } }{ 2 } = 5.172 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.58**2+2 * 9.11**2 - 8.6**2 } }{ 2 } = 5.788 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.6**2+2 * 9.11**2 - 4.58**2 } }{ 2 } = 8.559 ; ;
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