21 24 28 triangle

Acute scalene triangle.

Sides: a = 21   b = 24   c = 28

Area: T = 245.1755319924
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 46.86602822563° = 46°51'37″ = 0.81878662138 rad
Angle ∠ B = β = 56.50545509846° = 56°30'16″ = 0.9866190457 rad
Angle ∠ C = γ = 76.63551667591° = 76°38'7″ = 1.33875359828 rad

Height: ha = 23.3550030469
Height: hb = 20.43112766604
Height: hc = 17.51325228517

Median: ma = 23.86994365246
Median: mb = 21.64548608219
Median: mc = 17.67876695297

Inradius: r = 6.71771320527
Circumradius: R = 14.39897028506

Vertex coordinates: A[28; 0] B[0; 0] C[11.58992857143; 17.51325228517]
Centroid: CG[13.19664285714; 5.83875076172]
Coordinates of the circumscribed circle: U[14; 3.32661912343]
Coordinates of the inscribed circle: I[12.5; 6.71771320527]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.1439717744° = 133°8'23″ = 0.81878662138 rad
∠ B' = β' = 123.4955449015° = 123°29'44″ = 0.9866190457 rad
∠ C' = γ' = 103.3654833241° = 103°21'53″ = 1.33875359828 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 = 21 ; ; b = 24 ; ; c = 28 ; ;

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

p = a+b+c = 21+24+28 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-21)(36.5-24)(36.5-28) } ; ; T = sqrt{ 60110.94 } = 245.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 245.18 }{ 21 } = 23.35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 245.18 }{ 24 } = 20.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 245.18 }{ 28 } = 17.51 ; ;

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+28**2-21**2 }{ 2 * 24 * 28 } ) = 46° 51'37" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21**2+28**2-24**2 }{ 2 * 21 * 28 } ) = 56° 30'16" ; ; gamma = 180° - alpha - beta = 180° - 46° 51'37" - 56° 30'16" = 76° 38'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 245.18 }{ 36.5 } = 6.72 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 21 }{ 2 * sin 46° 51'37" } = 14.39 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 28**2 - 21**2 } }{ 2 } = 23.869 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 28**2+2 * 21**2 - 24**2 } }{ 2 } = 21.645 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 24**2+2 * 21**2 - 28**2 } }{ 2 } = 17.678 ; ;
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