10 16 23 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 16   c = 23

Area: T = 67.30110958306
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 21.45547820505° = 21°27'17″ = 0.37444565871 rad
Angle ∠ B = β = 35.8199022505° = 35°49'8″ = 0.62551598776 rad
Angle ∠ C = γ = 122.7266195444° = 122°43'34″ = 2.1421976189 rad

Height: ha = 13.46602191661
Height: hb = 8.41326369788
Height: hc = 5.85222692027

Median: ma = 19.17702895127
Median: mb = 15.82771917913
Median: mc = 6.76438746292

Inradius: r = 2.74769835033
Circumradius: R = 13.67699111455

Vertex coordinates: A[23; 0] B[0; 0] C[8.10986956522; 5.85222692027]
Centroid: CG[10.37695652174; 1.95107564009]
Coordinates of the circumscribed circle: U[11.5; -7.3990295713]
Coordinates of the inscribed circle: I[8.5; 2.74769835033]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.545521795° = 158°32'43″ = 0.37444565871 rad
∠ B' = β' = 144.1810977495° = 144°10'52″ = 0.62551598776 rad
∠ C' = γ' = 57.27438045555° = 57°16'26″ = 2.1421976189 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 = 10 ; ; b = 16 ; ; c = 23 ; ;

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

p = a+b+c = 10+16+23 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-10)(24.5-16)(24.5-23) } ; ; T = sqrt{ 4529.44 } = 67.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 67.3 }{ 10 } = 13.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 67.3 }{ 16 } = 8.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 67.3 }{ 23 } = 5.85 ; ;

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{ 10**2-16**2-23**2 }{ 2 * 16 * 23 } ) = 21° 27'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-10**2-23**2 }{ 2 * 10 * 23 } ) = 35° 49'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-10**2-16**2 }{ 2 * 16 * 10 } ) = 122° 43'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 67.3 }{ 24.5 } = 2.75 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 21° 27'17" } = 13.67 ; ;




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