Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 12.53299640861   b = 8.54440037453   c = 9.48768329805

Area: T = 40.5
Perimeter: p = 30.5610800812
Semiperimeter: s = 15.2880400406

Angle ∠ A = α = 87.87989036033° = 87°52'44″ = 1.53437762109 rad
Angle ∠ B = β = 42.95545915111° = 42°57'17″ = 0.75496990507 rad
Angle ∠ C = γ = 49.16765048855° = 49°9'59″ = 0.8588117392 rad

Height: ha = 6.46545037642
Height: hb = 9.48803329229
Height: hc = 8.53881496825

Median: ma = 6.5
Median: mb = 10.25991422643
Median: mc = 9.61876920308

Inradius: r = 2.65504541062
Circumradius: R = 6.26992775403

Vertex coordinates: A[5; 8; 0] B[-4; 11; 0] C[2; 0; 0]
Centroid: CG[1; 6.33333333333; 0]
Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 92.12110963967° = 92°7'16″ = 1.53437762109 rad
∠ B' = β' = 137.0455408489° = 137°2'43″ = 0.75496990507 rad
∠ C' = γ' = 130.8333495114° = 130°50'1″ = 0.8588117392 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{ (-4-2)**2 + (11-0)**2 } ; ; a = sqrt{ 157 } = 12.53 ; ;

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{ (5-2)**2 + (8-0)**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{ (5-(-4))**2 + (8-11)**2 } ; ; c = sqrt{ 90 } = 9.49 ; ;


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 = 12.53 ; ; b = 8.54 ; ; c = 9.49 ; ;

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

p = a+b+c = 12.53+8.54+9.49 = 30.56 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.56 }{ 2 } = 15.28 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.28 * (15.28-12.53)(15.28-8.54)(15.28-9.49) } ; ; T = sqrt{ 1640.25 } = 40.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.5 }{ 12.53 } = 6.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.5 }{ 8.54 } = 9.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.5 }{ 9.49 } = 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+9.49**2-12.53**2 }{ 2 * 8.54 * 9.49 } ) = 87° 52'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12.53**2+9.49**2-8.54**2 }{ 2 * 12.53 * 9.49 } ) = 42° 57'17" ; ; gamma = 180° - alpha - beta = 180° - 87° 52'44" - 42° 57'17" = 49° 9'59" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.5 }{ 15.28 } = 2.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12.53 }{ 2 * sin 87° 52'44" } = 6.27 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 9.49**2 - 12.53**2 } }{ 2 } = 6.5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.49**2+2 * 12.53**2 - 8.54**2 } }{ 2 } = 10.259 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.54**2+2 * 12.53**2 - 9.49**2 } }{ 2 } = 9.618 ; ;
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