Triangle calculator VC

Please enter the coordinates of the three vertices


Right isosceles triangle.

Sides: a = 4.4722135955   b = 6.32545553203   c = 4.4722135955

Area: T = 10
Perimeter: p = 15.26988272303
Semiperimeter: s = 7.63444136152

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 4.4722135955
Height: hb = 3.16222776602
Height: hc = 4.4722135955

Median: ma = 5
Median: mb = 3.16222776602
Median: mc = 5

Inradius: r = 1.31098582948
Circumradius: R = 3.16222776602

Vertex coordinates: A[-1; 3] B[3; 5] C[5; 1]
Centroid: CG[2.33333333333; 3]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[-0; 1.31098582948]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 135° = 0.78553981634 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{ (3-5)**2 + (5-1)**2 } ; ; a = sqrt{ 20 } = 4.47 ; ;

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{ (-1-5)**2 + (3-1)**2 } ; ; b = sqrt{ 40 } = 6.32 ; ;

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 + (3-5)**2 } ; ; c = sqrt{ 20 } = 4.47 ; ;


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 = 4.47 ; ; b = 6.32 ; ; c = 4.47 ; ;

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

p = a+b+c = 4.47+6.32+4.47 = 15.27 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.27 }{ 2 } = 7.63 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.63 * (7.63-4.47)(7.63-6.32)(7.63-4.47) } ; ; T = sqrt{ 100 } = 10 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10 }{ 4.47 } = 4.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10 }{ 6.32 } = 3.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10 }{ 4.47 } = 4.47 ; ;

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{ 6.32**2+4.47**2-4.47**2 }{ 2 * 6.32 * 4.47 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.47**2+4.47**2-6.32**2 }{ 2 * 4.47 * 4.47 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 45° - 90° = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10 }{ 7.63 } = 1.31 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.47 }{ 2 * sin 45° } = 3.16 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.32**2+2 * 4.47**2 - 4.47**2 } }{ 2 } = 5 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.47**2+2 * 4.47**2 - 6.32**2 } }{ 2 } = 3.162 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.32**2+2 * 4.47**2 - 4.47**2 } }{ 2 } = 5 ; ;
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