10 22 23 triangle

Acute scalene triangle.

Sides: a = 10   b = 22   c = 23

Area: T = 109.1377241581
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 25.55546937848° = 25°33'17″ = 0.44660135459 rad
Angle ∠ B = β = 71.62660619967° = 71°37'34″ = 1.25501106121 rad
Angle ∠ C = γ = 82.81992442185° = 82°49'9″ = 1.44554684956 rad

Height: ha = 21.82774483163
Height: hb = 9.92215674165
Height: hc = 9.49901949201

Median: ma = 21.94331082575
Median: mb = 13.91104277432
Median: mc = 12.63992246598

Inradius: r = 3.96986269666
Circumradius: R = 11.59109105056

Vertex coordinates: A[23; 0] B[0; 0] C[3.1522173913; 9.49901949201]
Centroid: CG[8.71773913043; 3.16333983067]
Coordinates of the circumscribed circle: U[11.5; 1.44988638132]
Coordinates of the inscribed circle: I[5.5; 3.96986269666]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.4455306215° = 154°26'43″ = 0.44660135459 rad
∠ B' = β' = 108.3743938003° = 108°22'26″ = 1.25501106121 rad
∠ C' = γ' = 97.18107557815° = 97°10'51″ = 1.44554684956 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 = 22 ; ; c = 23 ; ;

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

p = a+b+c = 10+22+23 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-10)(27.5-22)(27.5-23) } ; ; T = sqrt{ 11910.94 } = 109.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.14 }{ 10 } = 21.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.14 }{ 22 } = 9.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.14 }{ 23 } = 9.49 ; ;

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**2-22**2-23**2 }{ 2 * 22 * 23 } ) = 25° 33'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-10**2-23**2 }{ 2 * 10 * 23 } ) = 71° 37'34" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-10**2-22**2 }{ 2 * 22 * 10 } ) = 82° 49'9" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.14 }{ 27.5 } = 3.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 25° 33'17" } = 11.59 ; ;




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