9 12 13 triangle

Acute scalene triangle.

Sides: a = 9 b = 12 c = 13

Area: T = 52.15436192416
Perimeter: p = 34
Semiperimeter: s = 17

Angle ∠ A = α = 41.96218888284° = 41°57'43″ = 0.73223731204 rad
Angle ∠ B = β = 63.06442249308° = 63°3'51″ = 1.10106783653 rad
Angle ∠ C = γ = 74.97438862409° = 74°58'26″ = 1.30985411679 rad

Height: h_a = 11.59896931648
Height: h_b = 8.69222698736
Height: h_c = 8.02436337295

Median: m_a = 11.67326175299
Median: m_b = 9.43439811321
Median: m_c = 8.38215273071

Inradius: r = 3.06878599554
Circumradius: R = 6.73301177771

Vertex coordinates: A[13; 0] B[0; 0] C[4.07769230769; 8.02436337295]
Centroid: CG[5.69223076923; 2.67545445765]
Coordinates of the circumscribed circle: U[6.5; 1.74548453496]
Coordinates of the inscribed circle: I[5; 3.06878599554]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.0388111172° = 138°2'17″ = 0.73223731204 rad
∠ B' = β' = 116.9365775069° = 116°56'9″ = 1.10106783653 rad
∠ C' = γ' = 105.0266113759° = 105°1'34″ = 1.30985411679 rad

Calculate another triangle

How did we calculate this triangle?

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.

$a = 9 ; ; b = 12 ; ; c = 13 ; ;$

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

$p = a+b+c = 9+12+13 = 34 ; ;$

2. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 34 }{ 2 } = 17 ; ;$

3. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 17 * (17-9)(17-12)(17-13) } ; ; T = sqrt{ 2720 } = 52.15 ; ;$

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

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 52.15 }{ 9 } = 11.59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 52.15 }{ 12 } = 8.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 52.15 }{ 13 } = 8.02 ; ;$

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

$a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 9**2-12**2-13**2 }{ 2 * 12 * 13 } ) = 41° 57'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-9**2-13**2 }{ 2 * 9 * 13 } ) = 63° 3'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-9**2-12**2 }{ 2 * 12 * 9 } ) = 74° 58'26" ; ;$

6. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 52.15 }{ 17 } = 3.07 ; ;$

7. Circumradius

$R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 41° 57'43" } = 6.73 ; ;$

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