Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.28801098893   b = 9.22195444573   c = 8.6022325267

Area: T = 29.5
Perimeter: p = 25.10219796136
Semiperimeter: s = 12.55109898068

Angle ∠ A = α = 48.06664855011° = 48°3'59″ = 0.83989184319 rad
Angle ∠ B = β = 70.40877181089° = 70°24'28″ = 1.22988464998 rad
Angle ∠ C = γ = 61.52657963899° = 61°31'33″ = 1.07438277219 rad

Height: ha = 8.1044273273
Height: hb = 6.39994485057
Height: hc = 6.85986106859

Median: ma = 8.1399410298
Median: mb = 6.5
Median: mc = 7.10663352018

Inradius: r = 2.35504122347
Circumradius: R = 4.8933067988

Vertex coordinates: A[-7; 1] B[0; -4] C[2; 3]
Centroid: CG[-1.66766666667; 0]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.83765874056; 2.35504122347]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.9343514499° = 131°56'1″ = 0.83989184319 rad
∠ B' = β' = 109.5922281891° = 109°35'32″ = 1.22988464998 rad
∠ C' = γ' = 118.474420361° = 118°28'27″ = 1.07438277219 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{ (0-2)**2 + (-4-3)**2 } ; ; a = sqrt{ 53 } = 7.28 ; ;

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{ (-7-2)**2 + (1-3)**2 } ; ; b = sqrt{ 85 } = 9.22 ; ;

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{ (-7-0)**2 + (1-(-4))**2 } ; ; c = sqrt{ 74 } = 8.6 ; ;


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 = 7.28 ; ; b = 9.22 ; ; c = 8.6 ; ;

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

p = a+b+c = 7.28+9.22+8.6 = 25.1 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.1 }{ 2 } = 12.55 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.55 * (12.55-7.28)(12.55-9.22)(12.55-8.6) } ; ; T = sqrt{ 870.25 } = 29.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.5 }{ 7.28 } = 8.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.5 }{ 9.22 } = 6.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.5 }{ 8.6 } = 6.86 ; ;

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{ 9.22**2+8.6**2-7.28**2 }{ 2 * 9.22 * 8.6 } ) = 48° 3'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.28**2+8.6**2-9.22**2 }{ 2 * 7.28 * 8.6 } ) = 70° 24'28" ; ; gamma = 180° - alpha - beta = 180° - 48° 3'59" - 70° 24'28" = 61° 31'33" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.5 }{ 12.55 } = 2.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.28 }{ 2 * sin 48° 3'59" } = 4.89 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.22**2+2 * 8.6**2 - 7.28**2 } }{ 2 } = 8.139 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.6**2+2 * 7.28**2 - 9.22**2 } }{ 2 } = 6.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.22**2+2 * 7.28**2 - 8.6**2 } }{ 2 } = 7.106 ; ;
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