Triangle calculator VC

Please enter the coordinates of the three vertices


Obtuse scalene triangle.

Sides: a = 4.12331056256   b = 3.60655512755   c = 6.32545553203

Area: T = 7
Perimeter: p = 14.05332122214
Semiperimeter: s = 7.02766061107

Angle ∠ A = α = 37.87549836511° = 37°52'30″ = 0.66110431689 rad
Angle ∠ B = β = 32.47111922908° = 32°28'16″ = 0.56767292175 rad
Angle ∠ C = γ = 109.6543824058° = 109°39'14″ = 1.91438202672 rad

Height: ha = 3.39554987505
Height: hb = 3.88329013736
Height: hc = 2.21435943621

Median: ma = 4.7176990566
Median: mb = 5.02549378106
Median: mc = 2.23660679775

Inradius: r = 0.9966213519
Circumradius: R = 3.35879026496

Vertex coordinates: A[1; 4] B[3; -2] C[4; 2]
Centroid: CG[2.66766666667; 1.33333333333]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[1.5655478387; 0.9966213519]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.1255016349° = 142°7'30″ = 0.66110431689 rad
∠ B' = β' = 147.5298807709° = 147°31'44″ = 0.56767292175 rad
∠ C' = γ' = 70.34661759419° = 70°20'46″ = 1.91438202672 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{ (3-4)**2 + (-2-2)**2 } ; ; a = sqrt{ 17 } = 4.12 ; ;

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{ (1-4)**2 + (4-2)**2 } ; ; b = sqrt{ 13 } = 3.61 ; ;

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{ (1-3)**2 + (4-(-2))**2 } ; ; c = sqrt{ 40 } = 6.32 ; ;
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 = 4.12 ; ; b = 3.61 ; ; c = 6.32 ; ;

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

p = a+b+c = 4.12+3.61+6.32 = 14.05 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 14.05 }{ 2 } = 7.03 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.03 * (7.03-4.12)(7.03-3.61)(7.03-6.32) } ; ; T = sqrt{ 49 } = 7 ; ;

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 }{ 4.12 } = 3.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7 }{ 3.61 } = 3.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7 }{ 6.32 } = 2.21 ; ;

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{ 3.61**2+6.32**2-4.12**2 }{ 2 * 3.61 * 6.32 } ) = 37° 52'30" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.12**2+6.32**2-3.61**2 }{ 2 * 4.12 * 6.32 } ) = 32° 28'16" ; ; gamma = 180° - alpha - beta = 180° - 37° 52'30" - 32° 28'16" = 109° 39'14" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7 }{ 7.03 } = 1 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.12 }{ 2 * sin 37° 52'30" } = 3.36 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.61**2+2 * 6.32**2 - 4.12**2 } }{ 2 } = 4.717 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.32**2+2 * 4.12**2 - 3.61**2 } }{ 2 } = 5.025 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.61**2+2 * 4.12**2 - 6.32**2 } }{ 2 } = 2.236 ; ;
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