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: ha = 11.59896931648
Height: hb = 8.69222698736
Height: hc = 8.02436337295

Median: ma = 11.67326175299
Median: mb = 9.43439811321
Median: mc = 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

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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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 12**2+13**2-9**2 }{ 2 * 12 * 13 } ) = 41° 57'43" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9**2+13**2-12**2 }{ 2 * 9 * 13 } ) = 63° 3'51" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 9**2+12**2-13**2 }{ 2 * 9 * 12 } ) = 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 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 13**2 - 9**2 } }{ 2 } = 11.673 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 9**2 - 12**2 } }{ 2 } = 9.434 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 12**2+2 * 9**2 - 13**2 } }{ 2 } = 8.382 ; ;
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