20 24 24 triangle

Acute isosceles triangle.

Sides: a = 20   b = 24   c = 24

Area: T = 218.1744242293
Perimeter: p = 68
Semiperimeter: s = 34

Angle ∠ A = α = 49.24986367043° = 49°14'55″ = 0.86595508626 rad
Angle ∠ B = β = 65.37656816478° = 65°22'32″ = 1.14110208955 rad
Angle ∠ C = γ = 65.37656816478° = 65°22'32″ = 1.14110208955 rad

Height: ha = 21.81774242293
Height: hb = 18.18111868577
Height: hc = 18.18111868577

Median: ma = 21.81774242293
Median: mb = 18.5477236991
Median: mc = 18.5477236991

Inradius: r = 6.41768894792
Circumradius: R = 13.22004583572

Vertex coordinates: A[24; 0] B[0; 0] C[8.33333333333; 18.18111868577]
Centroid: CG[10.77877777778; 6.06603956192]
Coordinates of the circumscribed circle: U[12; 5.55001909822]
Coordinates of the inscribed circle: I[10; 6.41768894792]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.7511363296° = 130°45'5″ = 0.86595508626 rad
∠ B' = β' = 114.6244318352° = 114°37'28″ = 1.14110208955 rad
∠ C' = γ' = 114.6244318352° = 114°37'28″ = 1.14110208955 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 = 20 ; ; b = 24 ; ; c = 24 ; ;

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

p = a+b+c = 20+24+24 = 68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 68 }{ 2 } = 34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 34 * (34-20)(34-24)(34-24) } ; ; T = sqrt{ 47600 } = 218.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 218.17 }{ 20 } = 21.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 218.17 }{ 24 } = 18.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 218.17 }{ 24 } = 18.18 ; ;

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{ 24**2+24**2-20**2 }{ 2 * 24 * 24 } ) = 49° 14'55" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20**2+24**2-24**2 }{ 2 * 20 * 24 } ) = 65° 22'32" ; ; gamma = 180° - alpha - beta = 180° - 49° 14'55" - 65° 22'32" = 65° 22'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 218.17 }{ 34 } = 6.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20 }{ 2 * sin 49° 14'55" } = 13.2 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 24**2 - 20**2 } }{ 2 } = 21.817 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 20**2 - 24**2 } }{ 2 } = 18.547 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 20**2 - 24**2 } }{ 2 } = 18.547 ; ;
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