Triangle calculator VC

Please enter the coordinates of the three vertices


Acute scalene triangle.

Sides: a = 8.06222577483   b = 6.40331242374   c = 8.24662112512

Area: T = 24
Perimeter: p = 22.7121593237
Semiperimeter: s = 11.35657966185

Angle ∠ A = α = 65.37664352138° = 65°22'35″ = 1.14110340477 rad
Angle ∠ B = β = 46.21988752351° = 46°13'8″ = 0.80766715494 rad
Angle ∠ C = γ = 68.4054689551° = 68°24'17″ = 1.19438870565 rad

Height: ha = 5.95436672603
Height: hb = 7.49663405707
Height: hc = 5.82108550009

Median: ma = 6.18546584384
Median: mb = 7.5
Median: mc = 6

Inradius: r = 2.11334580696
Circumradius: R = 4.4344369005

Vertex coordinates: A[-2; 5] B[-4; -3] C[3; 1]
Centroid: CG[-1; 1]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[2.02553973167; 2.11334580696]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 114.6243564786° = 114°37'25″ = 1.14110340477 rad
∠ B' = β' = 133.7811124765° = 133°46'52″ = 0.80766715494 rad
∠ C' = γ' = 111.5955310449° = 111°35'43″ = 1.19438870565 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{ (-4-3)**2 + (-3-1)**2 } ; ; a = sqrt{ 65 } = 8.06 ; ;

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{ (-2-3)**2 + (5-1)**2 } ; ; b = sqrt{ 41 } = 6.4 ; ;

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{ (-2-(-4))**2 + (5-(-3))**2 } ; ; c = sqrt{ 68 } = 8.25 ; ;


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 = 8.06 ; ; b = 6.4 ; ; c = 8.25 ; ;

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

p = a+b+c = 8.06+6.4+8.25 = 22.71 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.71 }{ 2 } = 11.36 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.36 * (11.36-8.06)(11.36-6.4)(11.36-8.25) } ; ; T = sqrt{ 576 } = 24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24 }{ 8.06 } = 5.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24 }{ 6.4 } = 7.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24 }{ 8.25 } = 5.82 ; ;

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{ 6.4**2+8.25**2-8.06**2 }{ 2 * 6.4 * 8.25 } ) = 65° 22'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.06**2+8.25**2-6.4**2 }{ 2 * 8.06 * 8.25 } ) = 46° 13'8" ; ; gamma = 180° - alpha - beta = 180° - 65° 22'35" - 46° 13'8" = 68° 24'17" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24 }{ 11.36 } = 2.11 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.06 }{ 2 * sin 65° 22'35" } = 4.43 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.4**2+2 * 8.25**2 - 8.06**2 } }{ 2 } = 6.185 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.25**2+2 * 8.06**2 - 6.4**2 } }{ 2 } = 7.5 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.4**2+2 * 8.06**2 - 8.25**2 } }{ 2 } = 6 ; ;
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