Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 31.90661122671   b = 22.47222050542   c = 39.05112483795

Area: T = 358.5
Perimeter: p = 93.43295657009
Semiperimeter: s = 46.71547828504

Angle ∠ A = α = 54.7898682239° = 54°47'19″ = 0.95662428979 rad
Angle ∠ B = β = 35.13114073811° = 35°7'53″ = 0.61331587297 rad
Angle ∠ C = γ = 90.08799103799° = 90°4'48″ = 1.5722191026 rad

Height: ha = 22.47221831979
Height: hb = 31.90660812354
Height: hc = 18.36604885824

Median: ma = 27.57771644663
Median: mb = 33.8421542518
Median: mc = 19.5

Inradius: r = 7.6744230257
Circumradius: R = 19.52656431803

Vertex coordinates: A[-24; -19] B[-22; 20] C[-5; -7]
Centroid: CG[-17; -2]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[10.90766117599; 7.6744230257]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125.2111317761° = 125°12'41″ = 0.95662428979 rad
∠ B' = β' = 144.8698592619° = 144°52'7″ = 0.61331587297 rad
∠ C' = γ' = 89.92200896201° = 89°55'12″ = 1.5722191026 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{ (-22-(-5))**2 + (20-(-7))**2 } ; ; a = sqrt{ 1018 } = 31.91 ; ;

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{ (-24-(-5))**2 + (-19-(-7))**2 } ; ; b = sqrt{ 505 } = 22.47 ; ;

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{ (-24-(-22))**2 + (-19-20)**2 } ; ; c = sqrt{ 1525 } = 39.05 ; ;
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 = 31.91 ; ; b = 22.47 ; ; c = 39.05 ; ;

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

p = a+b+c = 31.91+22.47+39.05 = 93.43 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 93.43 }{ 2 } = 46.71 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 46.71 * (46.71-31.91)(46.71-22.47)(46.71-39.05) } ; ; T = sqrt{ 128522.25 } = 358.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 358.5 }{ 31.91 } = 22.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 358.5 }{ 22.47 } = 31.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 358.5 }{ 39.05 } = 18.36 ; ;

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{ 22.47**2+39.05**2-31.91**2 }{ 2 * 22.47 * 39.05 } ) = 54° 47'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 31.91**2+39.05**2-22.47**2 }{ 2 * 31.91 * 39.05 } ) = 35° 7'53" ; ;
 gamma = 180° - alpha - beta = 180° - 54° 47'19" - 35° 7'53" = 90° 4'48" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 358.5 }{ 46.71 } = 7.67 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 31.91 }{ 2 * sin 54° 47'19" } = 19.53 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.47**2+2 * 39.05**2 - 31.91**2 } }{ 2 } = 27.577 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.05**2+2 * 31.91**2 - 22.47**2 } }{ 2 } = 33.842 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.47**2+2 * 31.91**2 - 39.05**2 } }{ 2 } = 19.5 ; ;
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