Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 9.96989575884   b = 100.000015101   c = 1.60553220549

Area: T = 7.987672451
Perimeter: p = 21.57442947443
Semiperimeter: s = 10.78771473722

Angle ∠ A = α = 84.28548686825° = 84°17'6″ = 1.47110484681 rad
Angle ∠ B = β = 86.4954789245° = 86°29'41″ = 1.51096188581 rad
Angle ∠ C = γ = 9.22203420724° = 9°13'13″ = 0.16109253273 rad

Height: ha = 1.60223188862
Height: hb = 1.59773424899
Height: hc = 9.95503080837

Median: ma = 5.14223391185
Median: mb = 5.09769119715
Median: mc = 9.95221828754

Inradius: r = 0.7440392639
Circumradius: R = 5.00993788849

Vertex coordinates: A[23.4091; 103.1765] B[25; 102.9618] C[23.0652; 93.1824]
Centroid: CG[23.82547666667; 99.77435666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.04553519409; 0.7440392639]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 95.71551313175° = 95°42'54″ = 1.47110484681 rad
∠ B' = β' = 93.5055210755° = 93°30'19″ = 1.51096188581 rad
∠ C' = γ' = 170.7879657928° = 170°46'47″ = 0.16109253273 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{ (25-23.065)**2 + (102.962-93.182)**2 } ; ; a = sqrt{ 99.38 } = 9.97 ; ;

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{ (23.409-23.065)**2 + (103.177-93.182)**2 } ; ; b = sqrt{ 100 } = 10 ; ;

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{ (23.409-25)**2 + (103.177-102.962)**2 } ; ; c = sqrt{ 2.577 } = 1.61 ; ;


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 = 9.97 ; ; b = 10 ; ; c = 1.61 ; ;

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

p = a+b+c = 9.97+10+1.61 = 21.57 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21.57 }{ 2 } = 10.79 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.79 * (10.79-9.97)(10.79-10)(10.79-1.61) } ; ; T = sqrt{ 63.79 } = 7.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.99 }{ 9.97 } = 1.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.99 }{ 10 } = 1.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.99 }{ 1.61 } = 9.95 ; ;

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{ 9.97**2-10**2-1.61**2 }{ 2 * 10 * 1.61 } ) = 84° 17'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-9.97**2-1.61**2 }{ 2 * 9.97 * 1.61 } ) = 86° 29'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.61**2-9.97**2-10**2 }{ 2 * 10 * 9.97 } ) = 9° 13'13" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.99 }{ 10.79 } = 0.74 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9.97 }{ 2 * sin 84° 17'6" } = 5.01 ; ;




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