6 7 8 triangle

Acute scalene triangle.

Sides: a = 6   b = 7   c = 8

Area: T = 20.33331625676
Perimeter: p = 21
Semiperimeter: s = 10.5

Angle ∠ A = α = 46.56774634422° = 46°34'3″ = 0.81327555614 rad
Angle ∠ B = β = 57.91100487437° = 57°54'36″ = 1.01107210206 rad
Angle ∠ C = γ = 75.52224878141° = 75°31'21″ = 1.31881160717 rad

Height: ha = 6.77877208559
Height: hb = 5.80994750193
Height: hc = 5.08332906419

Median: ma = 6.8922024376
Median: mb = 6.14441028637
Median: mc = 5.14878150705

Inradius: r = 1.93664916731
Circumradius: R = 4.1311182236

Vertex coordinates: A[8; 0] B[0; 0] C[3.18875; 5.08332906419]
Centroid: CG[3.72991666667; 1.6944430214]
Coordinates of the circumscribed circle: U[4; 1.0332795559]
Coordinates of the inscribed circle: I[3.5; 1.93664916731]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.4332536558° = 133°25'57″ = 0.81327555614 rad
∠ B' = β' = 122.0989951256° = 122°5'24″ = 1.01107210206 rad
∠ C' = γ' = 104.4787512186° = 104°28'39″ = 1.31881160717 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 = 6 ; ; b = 7 ; ; c = 8 ; ;

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

p = a+b+c = 6+7+8 = 21 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 21 }{ 2 } = 10.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.5 * (10.5-6)(10.5-7)(10.5-8) } ; ; T = sqrt{ 413.44 } = 20.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.33 }{ 6 } = 6.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.33 }{ 7 } = 5.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.33 }{ 8 } = 5.08 ; ;

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{ 6**2-7**2-8**2 }{ 2 * 7 * 8 } ) = 46° 34'3" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7**2-6**2-8**2 }{ 2 * 6 * 8 } ) = 57° 54'36" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8**2-6**2-7**2 }{ 2 * 7 * 6 } ) = 75° 31'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.33 }{ 10.5 } = 1.94 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 46° 34'3" } = 4.13 ; ;




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