24 30 30 triangle

Acute isosceles triangle.

Sides: a = 24   b = 30   c = 30

Area: T = 329.9455450037
Perimeter: p = 84
Semiperimeter: s = 42

Angle ∠ A = α = 47.15663569564° = 47°9'23″ = 0.82330336921 rad
Angle ∠ B = β = 66.42218215218° = 66°25'19″ = 1.15992794807 rad
Angle ∠ C = γ = 66.42218215218° = 66°25'19″ = 1.15992794807 rad

Height: ha = 27.49554541697
Height: hb = 21.99663633358
Height: hc = 21.99663633358

Median: ma = 27.49554541697
Median: mb = 22.65495033058
Median: mc = 22.65495033058

Inradius: r = 7.85658440485
Circumradius: R = 16.36663417677

Vertex coordinates: A[30; 0] B[0; 0] C[9.6; 21.99663633358]
Centroid: CG[13.2; 7.33221211119]
Coordinates of the circumscribed circle: U[15; 6.54765367071]
Coordinates of the inscribed circle: I[12; 7.85658440485]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.8443643044° = 132°50'37″ = 0.82330336921 rad
∠ B' = β' = 113.5788178478° = 113°34'41″ = 1.15992794807 rad
∠ C' = γ' = 113.5788178478° = 113°34'41″ = 1.15992794807 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 = 24 ; ; b = 30 ; ; c = 30 ; ;

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

p = a+b+c = 24+30+30 = 84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 84 }{ 2 } = 42 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 42 * (42-24)(42-30)(42-30) } ; ; T = sqrt{ 108864 } = 329.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 329.95 }{ 24 } = 27.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 329.95 }{ 30 } = 22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 329.95 }{ 30 } = 22 ; ;

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{ 30**2+30**2-24**2 }{ 2 * 30 * 30 } ) = 47° 9'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 24**2+30**2-30**2 }{ 2 * 24 * 30 } ) = 66° 25'19" ; ; gamma = 180° - alpha - beta = 180° - 47° 9'23" - 66° 25'19" = 66° 25'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 329.95 }{ 42 } = 7.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 24 }{ 2 * sin 47° 9'23" } = 16.37 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 30**2 - 24**2 } }{ 2 } = 27.495 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 24**2 - 30**2 } }{ 2 } = 22.65 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 30**2+2 * 24**2 - 30**2 } }{ 2 } = 22.65 ; ;
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