3 8 10 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 8   c = 10

Area: T = 9.92215674165
Perimeter: p = 21
Semiperimeter: s = 10.5

Angle ∠ A = α = 14.36215115629° = 14°21'41″ = 0.25106556623 rad
Angle ∠ B = β = 41.41096221093° = 41°24'35″ = 0.72327342478 rad
Angle ∠ C = γ = 124.2298866328° = 124°13'44″ = 2.16882027434 rad

Height: ha = 6.61443782777
Height: hb = 2.48803918541
Height: hc = 1.98443134833

Median: ma = 8.93302855497
Median: mb = 6.2054836823
Median: mc = 3.39111649916

Inradius: r = 0.94549111825
Circumradius: R = 6.04774315681

Vertex coordinates: A[10; 0] B[0; 0] C[2.25; 1.98443134833]
Centroid: CG[4.08333333333; 0.66114378278]
Coordinates of the circumscribed circle: U[5; -3.40216802571]
Coordinates of the inscribed circle: I[2.5; 0.94549111825]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.6388488437° = 165°38'19″ = 0.25106556623 rad
∠ B' = β' = 138.5990377891° = 138°35'25″ = 0.72327342478 rad
∠ C' = γ' = 55.77111336722° = 55°46'16″ = 2.16882027434 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 = 8 ; ; c = 10 ; ;

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

p = a+b+c = 3+8+10 = 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 * (10.5-3)(10.5-8)(10.5-10) } ; ; T = sqrt{ 98.44 } = 9.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9.92 }{ 3 } = 6.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9.92 }{ 8 } = 2.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9.92 }{ 10 } = 1.98 ; ;

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**2-8**2-10**2 }{ 2 * 8 * 10 } ) = 14° 21'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8**2-3**2-10**2 }{ 2 * 3 * 10 } ) = 41° 24'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10**2-3**2-8**2 }{ 2 * 8 * 3 } ) = 124° 13'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9.92 }{ 10.5 } = 0.94 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 14° 21'41" } = 6.05 ; ;




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