Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 10.44403065089   b = 8.54440037453   c = 6.32545553203

Area: T = 27
Perimeter: p = 25.30988655746
Semiperimeter: s = 12.65444327873

Angle ∠ A = α = 87.87989036033° = 87°52'44″ = 1.53437762109 rad
Angle ∠ B = β = 54.86658069431° = 54°51'57″ = 0.95875889779 rad
Angle ∠ C = γ = 37.25552894536° = 37°15'19″ = 0.65502274647 rad

Height: ha = 5.17222619402
Height: hb = 6.32202219486
Height: hc = 8.53881496825

Median: ma = 5.40883269132
Median: mb = 7.5
Median: mc = 9

Inradius: r = 2.13436396861
Circumradius: R = 5.22437323795

Vertex coordinates: A[-9; -6] B[-3; -8] C[-6; 2]
Centroid: CG[-6; -4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.50114501495; 2.13436396861]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 92.12110963967° = 92°7'16″ = 1.53437762109 rad
∠ B' = β' = 125.1344193057° = 125°8'3″ = 0.95875889779 rad
∠ C' = γ' = 142.7454710546° = 142°44'41″ = 0.65502274647 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{ (-3-(-6))**2 + (-8-2)**2 } ; ; a = sqrt{ 109 } = 10.44 ; ;

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{ (-9-(-6))**2 + (-6-2)**2 } ; ; b = sqrt{ 73 } = 8.54 ; ;

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{ (-9-(-3))**2 + (-6-(-8))**2 } ; ; c = sqrt{ 40 } = 6.32 ; ;
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 = 10.44 ; ; b = 8.54 ; ; c = 6.32 ; ;

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

p = a+b+c = 10.44+8.54+6.32 = 25.31 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.31 }{ 2 } = 12.65 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.65 * (12.65-10.44)(12.65-8.54)(12.65-6.32) } ; ; T = sqrt{ 729 } = 27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27 }{ 10.44 } = 5.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27 }{ 8.54 } = 6.32 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27 }{ 6.32 } = 8.54 ; ;

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{ 8.54**2+6.32**2-10.44**2 }{ 2 * 8.54 * 6.32 } ) = 87° 52'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.44**2+6.32**2-8.54**2 }{ 2 * 10.44 * 6.32 } ) = 54° 51'57" ; ; gamma = 180° - alpha - beta = 180° - 87° 52'44" - 54° 51'57" = 37° 15'19" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27 }{ 12.65 } = 2.13 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.44 }{ 2 * sin 87° 52'44" } = 5.22 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 6.32**2 - 10.44**2 } }{ 2 } = 5.408 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.32**2+2 * 10.44**2 - 8.54**2 } }{ 2 } = 7.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 10.44**2 - 6.32**2 } }{ 2 } = 9 ; ;
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