Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 14.86660687473   b = 13.03884048104   c = 6.08327625303

Area: T = 39.5
Perimeter: p = 33.9877236088
Semiperimeter: s = 16.9943618044

Angle ∠ A = α = 95.0643616853° = 95°3'49″ = 1.65991731129 rad
Angle ∠ B = β = 60.88438537339° = 60°53'2″ = 1.0632623709 rad
Angle ∠ C = γ = 24.0532529413° = 24°3'9″ = 0.42197958317 rad

Height: ha = 5.31441150726
Height: hb = 6.05990234119
Height: hc = 12.98875199971

Median: ma = 6.94662219947
Median: mb = 9.30105376189
Median: mc = 13.6477344064

Inradius: r = 2.32444020136
Circumradius: R = 7.46221568363

Vertex coordinates: A[0; -8] B[6; -9] C[1; 5]
Centroid: CG[2.33333333333; -4]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.29546036531; 2.32444020136]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 84.9366383147° = 84°56'11″ = 1.65991731129 rad
∠ B' = β' = 119.11661462661° = 119°6'58″ = 1.0632623709 rad
∠ C' = γ' = 155.9477470587° = 155°56'51″ = 0.42197958317 rad

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How did we calculate this triangle?

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

a = |BC| = | beta - gamma | ; ; 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{ (6-1)**2 + (-9-5)**2 } ; ; a = sqrt{ 221 } = 14.87 ; ;

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

b = |AC| = | alpha - gamma | ; ; 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{ (0-1)**2 + (-8-5)**2 } ; ; b = sqrt{ 170 } = 13.04 ; ;

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

c = |AB| = | alpha - beta | ; ; 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{ (0-6)**2 + (-8-(-9))**2 } ; ; c = sqrt{ 37 } = 6.08 ; ;


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 = 14.87 ; ; b = 13.04 ; ; c = 6.08 ; ;

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

p = a+b+c = 14.87+13.04+6.08 = 33.99 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 33.99 }{ 2 } = 16.99 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.99 * (16.99-14.87)(16.99-13.04)(16.99-6.08) } ; ; T = sqrt{ 1560.25 } = 39.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 39.5 }{ 14.87 } = 5.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 39.5 }{ 13.04 } = 6.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 39.5 }{ 6.08 } = 12.99 ; ;

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{ 13.04**2+6.08**2-14.87**2 }{ 2 * 13.04 * 6.08 } ) = 95° 3'49" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 14.87**2+6.08**2-13.04**2 }{ 2 * 14.87 * 6.08 } ) = 60° 53'2" ; ; gamma = 180° - alpha - beta = 180° - 95° 3'49" - 60° 53'2" = 24° 3'9" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 39.5 }{ 16.99 } = 2.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14.87 }{ 2 * sin 95° 3'49" } = 7.46 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.04**2+2 * 6.08**2 - 14.87**2 } }{ 2 } = 6.946 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.08**2+2 * 14.87**2 - 13.04**2 } }{ 2 } = 9.301 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13.04**2+2 * 14.87**2 - 6.08**2 } }{ 2 } = 13.647 ; ;
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