Triangle calculator VC

Acute scalene triangle.

Sides: a = 7.21111025509 b = 12.64991106407 c = 12.80662484749

Area: T = 44
Perimeter: p = 32.66664616665
Semiperimeter: s = 16.33332308332

Angle ∠ A = α = 32.9055242923° = 32°54'19″ = 0.57443048302 rad
Angle ∠ B = β = 72.35498757801° = 72°21' = 1.26327435458 rad
Angle ∠ C = γ = 74.74548812969° = 74°44'42″ = 1.30545442776 rad

Height: h_a = 12.2033404317
Height: h_b = 6.95770108524
Height: h_c = 6.87216455231

Median: m_a = 12.20765556157
Median: m_b = 8.24662112512
Median: m_c = 8.06222577483

Inradius: r = 2.69438944566
Circumradius: R = 6.63769862722

Vertex coordinates: A[2; 0] B[-6; 10] C[-10; 4]
Centroid: CG[-4.66766666667; 4.66766666667]
Coordinates of the circumscribed circle: U[0; 0]
Coordinates of the inscribed circle: I[0.85771482362; 2.69438944566]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.0954757077° = 147°5'41″ = 0.57443048302 rad
∠ B' = β' = 107.655012422° = 107°39' = 1.26327435458 rad
∠ C' = γ' = 105.2555118703° = 105°15'18″ = 1.30545442776 rad

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How did we calculate this triangle?

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

$a = |BC| = |B-C| ; ; a**2 = (B_x-C_x)**2 + (B_y-C_y)**2 ; ; a = sqrt{ (B_x-C_x)**2 + (B_y-C_y)**2 } ; ; a = sqrt{ (-6-(-10))**2 + (10-4)**2 } ; ; a = sqrt{ 52 } = 7.21 ; ;$

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

$b = |AC| = |A-C| ; ; b**2 = (A_x-C_x)**2 + (A_y-C_y)**2 ; ; b = sqrt{ (A_x-C_x)**2 + (A_y-C_y)**2 } ; ; b = sqrt{ (2-(-10))**2 + (0-4)**2 } ; ; b = sqrt{ 160 } = 12.65 ; ;$

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

$c = |AB| = |A-B| ; ; c**2 = (A_x-B_x)**2 + (A_y-B_y)**2 ; ; c = sqrt{ (A_x-B_x)**2 + (A_y-B_y)**2 } ; ; c = sqrt{ (2-(-6))**2 + (0-10)**2 } ; ; c = sqrt{ 164 } = 12.81 ; ;$
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.

$a = 7.21 ; ; b = 12.65 ; ; c = 12.81 ; ;$

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

$p = a+b+c = 7.21+12.65+12.81 = 32.67 ; ;$

5. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 32.67 }{ 2 } = 16.33 ; ;$

6. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.33 * (16.33-7.21)(16.33-12.65)(16.33-12.81) } ; ; T = sqrt{ 1936 } = 44 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 44 }{ 7.21 } = 12.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 44 }{ 12.65 } = 6.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 44 }{ 12.81 } = 6.87 ; ;$

8. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12.65**2+12.81**2-7.21**2 }{ 2 * 12.65 * 12.81 } ) = 32° 54'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.21**2+12.81**2-12.65**2 }{ 2 * 7.21 * 12.81 } ) = 72° 21' ; ; gamma = 180° - alpha - beta = 180° - 32° 54'19" - 72° 21' = 74° 44'42" ; ;$

9. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 44 }{ 16.33 } = 2.69 ; ;$

10. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.21 }{ 2 * sin 32° 54'19" } = 6.64 ; ;$

11. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.65**2+2 * 12.81**2 - 7.21**2 } }{ 2 } = 12.207 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.81**2+2 * 7.21**2 - 12.65**2 } }{ 2 } = 8.246 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.65**2+2 * 7.21**2 - 12.81**2 } }{ 2 } = 8.062 ; ;$
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