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: h_a = 23.3550030469
Height: h_b = 20.43112766604
Height: h_c = 17.51325228517

Median: m_a = 23.86994365246
Median: m_b = 21.64548608219
Median: m_c = 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

Calculate another triangle

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

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

6. Inradius

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

7. Circumradius

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

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