Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 15.62204993518   b = 16.12545154966   c = 5.65768542495

Area: T = 44
Perimeter: p = 37.40218690979
Semiperimeter: s = 18.7010934549

Angle ∠ A = α = 74.74548812969° = 74°44'42″ = 1.30545442776 rad
Angle ∠ B = β = 84.80655710923° = 84°48'20″ = 1.48801364396 rad
Angle ∠ C = γ = 20.45495476108° = 20°26'58″ = 0.35769119364 rad

Height: ha = 5.6343622717
Height: hb = 5.45875283219
Height: hc = 15.55663491861

Median: ma = 9.22195444573
Median: mb = 8.54440037453
Median: mc = 15.62204993518

Inradius: r = 2.3532823592
Circumradius: R = 8.09655043131

Vertex coordinates: A[-5; 2] B[-1; 6] C[9; -6]
Centroid: CG[1; 0.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.21438930538; 2.3532823592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.2555118703° = 105°15'18″ = 1.30545442776 rad
∠ B' = β' = 95.19444289077° = 95°11'40″ = 1.48801364396 rad
∠ C' = γ' = 159.5550452389° = 159°33'2″ = 0.35769119364 rad

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How did we calculate this triangle?

1. We compute side a from coordinates using the Pythagorean theorem

a = | beta gamma | = | beta - gamma | ; ; a**2 = ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 ; ; a = sqrt{ ( beta _x- gamma _x)**2 + ( beta _y- gamma _y)**2 } ; ; a = sqrt{ (-1-9)**2 + (6-(-6))**2 } ; ; a = sqrt{ 244 } = 15.62 ; ;

2. We compute side b from coordinates using the Pythagorean theorem

b = | alpha gamma | = | alpha - gamma | ; ; b**2 = ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 ; ; b = sqrt{ ( alpha _x- gamma _x)**2 + ( alpha _y- gamma _y)**2 } ; ; b = sqrt{ (-5-9)**2 + (2-(-6))**2 } ; ; b = sqrt{ 260 } = 16.12 ; ;

3. We compute side c from coordinates using the Pythagorean theorem

c = | alpha beta | = | alpha - beta | ; ; c**2 = ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 ; ; c = sqrt{ ( alpha _x- beta _x)**2 + ( alpha _y- beta _y)**2 } ; ; c = sqrt{ (-5-(-1))**2 + (2-6)**2 } ; ; c = sqrt{ 32 } = 5.66 ; ;


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 = 15.62 ; ; b = 16.12 ; ; c = 5.66 ; ;

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

p = a+b+c = 15.62+16.12+5.66 = 37.4 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.4 }{ 2 } = 18.7 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.7 * (18.7-15.62)(18.7-16.12)(18.7-5.66) } ; ; T = sqrt{ 1936 } = 44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44 }{ 15.62 } = 5.63 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44 }{ 16.12 } = 5.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44 }{ 5.66 } = 15.56 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 15.62**2-16.12**2-5.66**2 }{ 2 * 16.12 * 5.66 } ) = 74° 44'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16.12**2-15.62**2-5.66**2 }{ 2 * 15.62 * 5.66 } ) = 84° 48'20" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.66**2-15.62**2-16.12**2 }{ 2 * 16.12 * 15.62 } ) = 20° 26'58" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 44 }{ 18.7 } = 2.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15.62 }{ 2 * sin 74° 44'42" } = 8.1 ; ;




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