Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 18.43990889146   b = 2.23660679775   c = 19.92548588452

Area: T = 16
Perimeter: p = 40.66000157373
Semiperimeter: s = 20.33000078686

Angle ∠ A = α = 45.90993804492° = 45°54'34″ = 0.80112698464 rad
Angle ∠ B = β = 4.99767606646° = 4°59'48″ = 0.08772099255 rad
Angle ∠ C = γ = 129.0943858886° = 129°5'38″ = 2.25331128817 rad

Height: ha = 1.73554436625
Height: hb = 14.3110835056
Height: hc = 1.60660339623

Median: ma = 10.77703296143
Median: mb = 19.16437678967
Median: mc = 8.55986213843

Inradius: r = 0.78881770344
Circumradius: R = 12.83662964991

Vertex coordinates: A[5; -10] B[-1; 9] C[3; -9]
Centroid: CG[2.33333333333; -3.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[9.01547748308; 0.78881770344]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.0910619551° = 134°5'26″ = 0.80112698464 rad
∠ B' = β' = 175.0033239335° = 175°12″ = 0.08772099255 rad
∠ C' = γ' = 50.90661411138° = 50°54'22″ = 2.25331128817 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{ (-1-3)**2 + (9-(-9))**2 } ; ; a = sqrt{ 340 } = 18.44 ; ;

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{ (5-3)**2 + (-10-(-9))**2 } ; ; b = sqrt{ 5 } = 2.24 ; ;

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{ (5-(-1))**2 + (-10-9)**2 } ; ; c = sqrt{ 397 } = 19.92 ; ;
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 = 18.44 ; ; b = 2.24 ; ; c = 19.92 ; ;

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

p = a+b+c = 18.44+2.24+19.92 = 40.6 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.6 }{ 2 } = 20.3 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.3 * (20.3-18.44)(20.3-2.24)(20.3-19.92) } ; ; T = sqrt{ 256 } = 16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 16 }{ 18.44 } = 1.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 16 }{ 2.24 } = 14.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 16 }{ 19.92 } = 1.61 ; ;

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{ 2.24**2+19.92**2-18.44**2 }{ 2 * 2.24 * 19.92 } ) = 45° 54'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 18.44**2+19.92**2-2.24**2 }{ 2 * 18.44 * 19.92 } ) = 4° 59'48" ; ;
 gamma = 180° - alpha - beta = 180° - 45° 54'34" - 4° 59'48" = 129° 5'38" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 16 }{ 20.3 } = 0.79 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 18.44 }{ 2 * sin 45° 54'34" } = 12.84 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 19.92**2 - 18.44**2 } }{ 2 } = 10.77 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.92**2+2 * 18.44**2 - 2.24**2 } }{ 2 } = 19.164 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.24**2+2 * 18.44**2 - 19.92**2 } }{ 2 } = 8.559 ; ;
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