Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 7.07110678119   b = 6.08327625303   c = 7.81102496759

Area: T = 20.5
Perimeter: p = 20.96440800181
Semiperimeter: s = 10.4822040009

Angle ∠ A = α = 59.65767511158° = 59°39'24″ = 1.0411206728 rad
Angle ∠ B = β = 47.93656734464° = 47°56'8″ = 0.83766353308 rad
Angle ∠ C = γ = 72.40875754378° = 72°24'27″ = 1.26437505948 rad

Height: ha = 5.79882756057
Height: hb = 6.74403584795
Height: hc = 5.25495120772

Median: ma = 6.04215229868
Median: mb = 6.80107352544
Median: mc = 5.31550729064

Inradius: r = 1.95657261738
Circumradius: R = 4.09767261054

Vertex coordinates: A[7; 6] B[1; 1] C[8; 0]
Centroid: CG[5.33333333333; 2.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.76549236202; 1.95657261738]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.3433248884° = 120°20'36″ = 1.0411206728 rad
∠ B' = β' = 132.0644326554° = 132°3'52″ = 0.83766353308 rad
∠ C' = γ' = 107.5922424562° = 107°35'33″ = 1.26437505948 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{ (1-8)**2 + (1-0)**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{ (7-8)**2 + (6-0)**2 } ; ; b = sqrt{ 37 } = 6.08 ; ;

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{ (7-1)**2 + (6-1)**2 } ; ; c = sqrt{ 61 } = 7.81 ; ;
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 = 6.08 ; ; c = 7.81 ; ;

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

p = a+b+c = 7.07+6.08+7.81 = 20.96 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.96 }{ 2 } = 10.48 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.48 * (10.48-7.07)(10.48-6.08)(10.48-7.81) } ; ; T = sqrt{ 420.25 } = 20.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.5 }{ 7.07 } = 5.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.5 }{ 6.08 } = 6.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.5 }{ 7.81 } = 5.25 ; ;

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{ 6.08**2+7.81**2-7.07**2 }{ 2 * 6.08 * 7.81 } ) = 59° 39'24" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.07**2+7.81**2-6.08**2 }{ 2 * 7.07 * 7.81 } ) = 47° 56'8" ; ; gamma = 180° - alpha - beta = 180° - 59° 39'24" - 47° 56'8" = 72° 24'27" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.5 }{ 10.48 } = 1.96 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.07 }{ 2 * sin 59° 39'24" } = 4.1 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.08**2+2 * 7.81**2 - 7.07**2 } }{ 2 } = 6.042 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 7.07**2 - 6.08**2 } }{ 2 } = 6.801 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.08**2+2 * 7.07**2 - 7.81**2 } }{ 2 } = 5.315 ; ;
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