7 8 13 triangle

Obtuse scalene triangle.

Sides: a = 7   b = 8   c = 13

Area: T = 24.2498711306
Perimeter: p = 28
Semiperimeter: s = 14

Angle ∠ A = α = 27.7965772496° = 27°47'45″ = 0.48551277482 rad
Angle ∠ B = β = 32.2044227504° = 32°12'15″ = 0.5622069803 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 6.92882032303
Height: hb = 6.06221778265
Height: hc = 3.73105709701

Median: ma = 10.21102889283
Median: mb = 9.6443650761
Median: mc = 3.77549172176

Inradius: r = 1.73220508076
Circumradius: R = 7.50655534995

Vertex coordinates: A[13; 0] B[0; 0] C[5.92330769231; 3.73105709701]
Centroid: CG[6.30876923077; 1.24435236567]
Coordinates of the circumscribed circle: U[6.5; -3.75327767497]
Coordinates of the inscribed circle: I[6; 1.73220508076]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.2044227504° = 152°12'15″ = 0.48551277482 rad
∠ B' = β' = 147.7965772496° = 147°47'45″ = 0.5622069803 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 = 7 ; ; b = 8 ; ; c = 13 ; ;

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

p = a+b+c = 7+8+13 = 28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28 }{ 2 } = 14 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14 * (14-7)(14-8)(14-13) } ; ; T = sqrt{ 588 } = 24.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24.25 }{ 7 } = 6.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24.25 }{ 8 } = 6.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24.25 }{ 13 } = 3.73 ; ;

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{ 7**2-8**2-13**2 }{ 2 * 8 * 13 } ) = 27° 47'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8**2-7**2-13**2 }{ 2 * 7 * 13 } ) = 32° 12'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-7**2-8**2 }{ 2 * 8 * 7 } ) = 120° ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24.25 }{ 14 } = 1.73 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 27° 47'45" } = 7.51 ; ;




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