5 8 9 triangle

Acute scalene triangle.

Sides: a = 5   b = 8   c = 9

Area: T = 19.98997487421
Perimeter: p = 22
Semiperimeter: s = 11

Angle ∠ A = α = 33.55773097619° = 33°33'26″ = 0.58656855435 rad
Angle ∠ B = β = 62.18218607153° = 62°10'55″ = 1.08552782045 rad
Angle ∠ C = γ = 84.26108295227° = 84°15'39″ = 1.47106289056 rad

Height: ha = 7.96598994969
Height: hb = 4.97549371855
Height: hc = 4.42221663871

Median: ma = 8.1399410298
Median: mb = 6.08327625303
Median: mc = 4.92444289009

Inradius: r = 1.80990680675
Circumradius: R = 4.52326701687

Vertex coordinates: A[9; 0] B[0; 0] C[2.33333333333; 4.42221663871]
Centroid: CG[3.77877777778; 1.47440554624]
Coordinates of the circumscribed circle: U[4.5; 0.45222670169]
Coordinates of the inscribed circle: I[3; 1.80990680675]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad
∠ B' = β' = 117.8188139285° = 117°49'5″ = 1.08552782045 rad
∠ C' = γ' = 95.73991704773° = 95°44'21″ = 1.47106289056 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 = 5 ; ; b = 8 ; ; c = 9 ; ;

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

p = a+b+c = 5+8+9 = 22 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22 }{ 2 } = 11 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11 * (11-5)(11-8)(11-9) } ; ; T = sqrt{ 396 } = 19.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19.9 }{ 5 } = 7.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19.9 }{ 8 } = 4.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19.9 }{ 9 } = 4.42 ; ;

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{ 8**2+9**2-5**2 }{ 2 * 8 * 9 } ) = 33° 33'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+9**2-8**2 }{ 2 * 5 * 9 } ) = 62° 10'55" ; ; gamma = 180° - alpha - beta = 180° - 33° 33'26" - 62° 10'55" = 84° 15'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19.9 }{ 11 } = 1.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 33° 33'26" } = 4.52 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 9**2 - 5**2 } }{ 2 } = 8.139 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9**2+2 * 5**2 - 8**2 } }{ 2 } = 6.083 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 5**2 - 9**2 } }{ 2 } = 4.924 ; ;
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