5 7 8 triangle

Acute scalene triangle.

Sides: a = 5   b = 7   c = 8

Area: T = 17.32105080757
Perimeter: p = 20
Semiperimeter: s = 10

Angle ∠ A = α = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 81.78767892983° = 81°47'12″ = 1.42774487579 rad

Height: ha = 6.92882032303
Height: hb = 4.94987165931
Height: hc = 4.33301270189

Median: ma = 7.08987234394
Median: mb = 5.67989083458
Median: mc = 4.5832575695

Inradius: r = 1.73220508076
Circumradius: R = 4.04114518843

Vertex coordinates: A[8; 0] B[0; 0] C[2.5; 4.33301270189]
Centroid: CG[3.5; 1.4433375673]
Coordinates of the circumscribed circle: U[4; 0.57773502692]
Coordinates of the inscribed circle: I[3; 1.73220508076]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 98.21332107017° = 98°12'48″ = 1.42774487579 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 = 7 ; ; c = 8 ; ;

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

p = a+b+c = 5+7+8 = 20 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20 }{ 2 } = 10 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10 * (10-5)(10-7)(10-8) } ; ; T = sqrt{ 300 } = 17.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.32 }{ 5 } = 6.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.32 }{ 7 } = 4.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.32 }{ 8 } = 4.33 ; ;

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{ 5**2-7**2-8**2 }{ 2 * 7 * 8 } ) = 38° 12'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7**2-5**2-8**2 }{ 2 * 5 * 8 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8**2-5**2-7**2 }{ 2 * 7 * 5 } ) = 81° 47'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.32 }{ 10 } = 1.73 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 38° 12'48" } = 4.04 ; ;




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