3 5 7 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 5   c = 7

Area: T = 6.49551905284
Perimeter: p = 15
Semiperimeter: s = 7.5

Angle ∠ A = α = 21.78767892983° = 21°47'12″ = 0.38802512067 rad
Angle ∠ B = β = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 4.33301270189
Height: hb = 2.59880762114
Height: hc = 1.85657687224

Median: ma = 5.89549130613
Median: mb = 4.77696960071
Median: mc = 2.17994494718

Inradius: r = 0.86660254038
Circumradius: R = 4.04114518843

Vertex coordinates: A[7; 0] B[0; 0] C[2.35771428571; 1.85657687224]
Centroid: CG[3.1199047619; 0.61985895741]
Coordinates of the circumscribed circle: U[3.5; -2.02107259422]
Coordinates of the inscribed circle: I[2.5; 0.86660254038]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2133210702° = 158°12'48″ = 0.38802512067 rad
∠ B' = β' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ C' = γ' = 60° = 2.09443951024 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 = 3 ; ; b = 5 ; ; c = 7 ; ;

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

p = a+b+c = 3+5+7 = 15 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15 }{ 2 } = 7.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.5 * (7.5-3)(7.5-5)(7.5-7) } ; ; T = sqrt{ 42.19 } = 6.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.5 }{ 3 } = 4.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.5 }{ 5 } = 2.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.5 }{ 7 } = 1.86 ; ;

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{ 3**2-5**2-7**2 }{ 2 * 5 * 7 } ) = 21° 47'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5**2-3**2-7**2 }{ 2 * 3 * 7 } ) = 38° 12'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 7**2-3**2-5**2 }{ 2 * 5 * 3 } ) = 120° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.5 }{ 7.5 } = 0.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 21° 47'12" } = 4.04 ; ;




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