5 6 7 triangle

Acute scalene triangle.

Sides: a = 5   b = 6   c = 7

Area: T = 14.69769384567
Perimeter: p = 18
Semiperimeter: s = 9

Angle ∠ A = α = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ B = β = 57.12216504356° = 57°7'18″ = 0.99769608743 rad
Angle ∠ C = γ = 78.46330409672° = 78°27'47″ = 1.3699438406 rad

Height: ha = 5.87987753827
Height: hb = 4.89989794856
Height: hc = 4.19991252733

Median: ma = 6.02107972894
Median: mb = 5.29215026221
Median: mc = 4.27220018727

Inradius: r = 1.63329931619
Circumradius: R = 3.57221725416

Vertex coordinates: A[7; 0] B[0; 0] C[2.71442857143; 4.19991252733]
Centroid: CG[3.23880952381; 1.43997084244]
Coordinates of the circumscribed circle: U[3.5; 0.71444345083]
Coordinates of the inscribed circle: I[3; 1.63329931619]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ B' = β' = 122.8788349564° = 122°52'42″ = 0.99769608743 rad
∠ C' = γ' = 101.5376959033° = 101°32'13″ = 1.3699438406 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 = 6 ; ; c = 7 ; ;

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

p = a+b+c = 5+6+7 = 18 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 18 }{ 2 } = 9 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9 * (9-5)(9-6)(9-7) } ; ; T = sqrt{ 216 } = 14.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14.7 }{ 5 } = 5.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14.7 }{ 6 } = 4.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14.7 }{ 7 } = 4.2 ; ;

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**2-6**2-7**2 }{ 2 * 6 * 7 } ) = 44° 24'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 6**2-5**2-7**2 }{ 2 * 5 * 7 } ) = 57° 7'18" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7**2-5**2-6**2 }{ 2 * 6 * 5 } ) = 78° 27'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14.7 }{ 9 } = 1.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 44° 24'55" } = 3.57 ; ;




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