5 8 10 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 8   c = 10

Area: T = 19.81100353357
Perimeter: p = 23
Semiperimeter: s = 11.5

Angle ∠ A = α = 29.68662952314° = 29°41'11″ = 0.51881235945 rad
Angle ∠ B = β = 52.41104970351° = 52°24'38″ = 0.91547357359 rad
Angle ∠ C = γ = 97.90332077335° = 97°54'12″ = 1.70987333232 rad

Height: ha = 7.92440141343
Height: hb = 4.95325088339
Height: hc = 3.96220070671

Median: ma = 8.70334475928
Median: mb = 6.81990908485
Median: mc = 4.41658804332

Inradius: r = 1.72326117683
Circumradius: R = 5.04879465738

Vertex coordinates: A[10; 0] B[0; 0] C[3.05; 3.96220070671]
Centroid: CG[4.35; 1.32106690224]
Coordinates of the circumscribed circle: U[5; -0.69440926539]
Coordinates of the inscribed circle: I[3.5; 1.72326117683]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.3143704769° = 150°18'49″ = 0.51881235945 rad
∠ B' = β' = 127.5989502965° = 127°35'22″ = 0.91547357359 rad
∠ C' = γ' = 82.09767922665° = 82°5'48″ = 1.70987333232 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 = 8 ; ; c = 10 ; ;

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

p = a+b+c = 5+8+10 = 23 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23 }{ 2 } = 11.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.5 * (11.5-5)(11.5-8)(11.5-10) } ; ; T = sqrt{ 392.44 } = 19.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 19.81 }{ 5 } = 7.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 19.81 }{ 8 } = 4.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 19.81 }{ 10 } = 3.96 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 8**2+10**2-5**2 }{ 2 * 8 * 10 } ) = 29° 41'11" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5**2+10**2-8**2 }{ 2 * 5 * 10 } ) = 52° 24'38" ; ; gamma = 180° - alpha - beta = 180° - 29° 41'11" - 52° 24'38" = 97° 54'12" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 19.81 }{ 11.5 } = 1.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5 }{ 2 * sin 29° 41'11" } = 5.05 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 10**2 - 5**2 } }{ 2 } = 8.703 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 5**2 - 8**2 } }{ 2 } = 6.819 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8**2+2 * 5**2 - 10**2 } }{ 2 } = 4.416 ; ;
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