Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.07110678119   b = 10.81766538264   c = 11.70546999107

Area: T = 37.5
Perimeter: p = 29.5922421549
Semiperimeter: s = 14.79662107745

Angle ∠ A = α = 36.32768259521° = 36°19'37″ = 0.63440227197 rad
Angle ∠ B = β = 64.98331065219° = 64°58'59″ = 1.1344169167 rad
Angle ∠ C = γ = 78.6990067526° = 78°41'24″ = 1.37334007669 rad

Height: ha = 10.60766017178
Height: hb = 6.93437524528
Height: hc = 6.40876824329

Median: ma = 10.77004672795
Median: mb = 8.01656097709
Median: mc = 7.01878344238

Inradius: r = 2.53444326714
Circumradius: R = 5.96882493246

Vertex coordinates: A[-5; 8] B[-1; -3] C[4; 2]
Centroid: CG[-0.66766666667; 2.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.18327352467; 2.53444326714]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143.6733174048° = 143°40'23″ = 0.63440227197 rad
∠ B' = β' = 115.0176893478° = 115°1'1″ = 1.1344169167 rad
∠ C' = γ' = 101.3109932474° = 101°18'36″ = 1.37334007669 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{ (-1-4)**2 + (-3-2)**2 } ; ; a = sqrt{ 50 } = 7.07 ; ;

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{ (-5-4)**2 + (8-2)**2 } ; ; b = sqrt{ 117 } = 10.82 ; ;

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-(-1))**2 + (8-(-3))**2 } ; ; c = sqrt{ 137 } = 11.7 ; ;


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 = 7.07 ; ; b = 10.82 ; ; c = 11.7 ; ;

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

p = a+b+c = 7.07+10.82+11.7 = 29.59 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.59 }{ 2 } = 14.8 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.8 * (14.8-7.07)(14.8-10.82)(14.8-11.7) } ; ; T = sqrt{ 1406.25 } = 37.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 37.5 }{ 7.07 } = 10.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 37.5 }{ 10.82 } = 6.93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 37.5 }{ 11.7 } = 6.41 ; ;

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{ 10.82**2+11.7**2-7.07**2 }{ 2 * 10.82 * 11.7 } ) = 36° 19'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.07**2+11.7**2-10.82**2 }{ 2 * 7.07 * 11.7 } ) = 64° 58'59" ; ; gamma = 180° - alpha - beta = 180° - 36° 19'37" - 64° 58'59" = 78° 41'24" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 37.5 }{ 14.8 } = 2.53 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.07 }{ 2 * sin 36° 19'37" } = 5.97 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.82**2+2 * 11.7**2 - 7.07**2 } }{ 2 } = 10.7 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.7**2+2 * 7.07**2 - 10.82**2 } }{ 2 } = 8.016 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.82**2+2 * 7.07**2 - 11.7**2 } }{ 2 } = 7.018 ; ;
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