Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 4.5832575695   b = 5.38551648071   c = 6.48107406984

Area: T = 12.17657956619
Perimeter: p = 16.44884812005
Semiperimeter: s = 8.22442406002

Angle ∠ A = α = 44.24772594687° = 44°14'50″ = 0.77222603627 rad
Angle ∠ B = β = 55.08108630005° = 55°4'51″ = 0.96113424142 rad
Angle ∠ C = γ = 80.67218775308° = 80°40'19″ = 1.40879898767 rad

Height: ha = 5.31439528826
Height: hb = 4.5221976986
Height: hc = 3.75875321182

Median: ma = 5.5
Median: mb = 4.92444289009
Median: mc = 3.80878865529

Inradius: r = 1.48804765879
Circumradius: R = 3.28437943339

Vertex coordinates: A[3; 0; 4] B[4; 5; 0] C[0; 4; 2]
Centroid: CG[2.33333333333; 3; 2]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.7532740531° = 135°45'10″ = 0.77222603627 rad
∠ B' = β' = 124.9199136999° = 124°55'9″ = 0.96113424142 rad
∠ C' = γ' = 99.32881224692° = 99°19'41″ = 1.40879898767 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{ (4-0)**2 + (5-4)**2 + (0 - 2)**2 } ; ; a = sqrt{ 21 } = 4.58 ; ;

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{ (3-0)**2 + (0-4)**2 + (4 - 2)**2 } ; ; b = sqrt{ 29 } = 5.39 ; ;

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{ (3-4)**2 + (0-5)**2 + (4 - 0)**2 } ; ; c = sqrt{ 42 } = 6.48 ; ;
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 = 4.58 ; ; b = 5.39 ; ; c = 6.48 ; ;

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

p = a+b+c = 4.58+5.39+6.48 = 16.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.45 }{ 2 } = 8.22 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.22 * (8.22-4.58)(8.22-5.39)(8.22-6.48) } ; ; T = sqrt{ 148.25 } = 12.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.18 }{ 4.58 } = 5.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.18 }{ 5.39 } = 4.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.18 }{ 6.48 } = 3.76 ; ;

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{ 5.39**2+6.48**2-4.58**2 }{ 2 * 5.39 * 6.48 } ) = 44° 14'50" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.58**2+6.48**2-5.39**2 }{ 2 * 4.58 * 6.48 } ) = 55° 4'51" ; ;
 gamma = 180° - alpha - beta = 180° - 44° 14'50" - 55° 4'51" = 80° 40'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.18 }{ 8.22 } = 1.48 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.58 }{ 2 * sin 44° 14'50" } = 3.28 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 6.48**2 - 4.58**2 } }{ 2 } = 5.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.48**2+2 * 4.58**2 - 5.39**2 } }{ 2 } = 4.924 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.39**2+2 * 4.58**2 - 6.48**2 } }{ 2 } = 3.808 ; ;
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