15 17 21 triangle

Acute scalene triangle.

Sides: a = 15   b = 17   c = 21

Area: T = 126.187711305
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 44.98657265175° = 44°59'9″ = 0.78551490441 rad
Angle ∠ B = β = 53.24437000714° = 53°14'37″ = 0.92992778722 rad
Angle ∠ C = γ = 81.77105734111° = 81°46'14″ = 1.42771657373 rad

Height: ha = 16.82549484067
Height: hb = 14.84655427118
Height: hc = 12.01878202905

Median: ma = 17.57112833908
Median: mb = 16.1487755262
Median: mc = 12.11440414396

Inradius: r = 4.7621777851
Circumradius: R = 10.60992450143

Vertex coordinates: A[21; 0] B[0; 0] C[8.97661904762; 12.01878202905]
Centroid: CG[9.99220634921; 4.00659400968]
Coordinates of the circumscribed circle: U[10.5; 1.51985782079]
Coordinates of the inscribed circle: I[9.5; 4.7621777851]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.0144273483° = 135°51″ = 0.78551490441 rad
∠ B' = β' = 126.7566299929° = 126°45'23″ = 0.92992778722 rad
∠ C' = γ' = 98.22994265889° = 98°13'46″ = 1.42771657373 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 = 15 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 15+17+21 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-15)(26.5-17)(26.5-21) } ; ; T = sqrt{ 15923.19 } = 126.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 126.19 }{ 15 } = 16.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 126.19 }{ 17 } = 14.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 126.19 }{ 21 } = 12.02 ; ;

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{ 15**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 44° 59'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 53° 14'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-15**2-17**2 }{ 2 * 17 * 15 } ) = 81° 46'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 126.19 }{ 26.5 } = 4.76 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 44° 59'9" } = 10.61 ; ;




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