15 21 21 triangle

Acute isosceles triangle.

Sides: a = 15   b = 21   c = 21

Area: T = 147.1132839345
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 41.85496648553° = 41°50'59″ = 0.73304144426 rad
Angle ∠ B = β = 69.07551675724° = 69°4'31″ = 1.20655891055 rad
Angle ∠ C = γ = 69.07551675724° = 69°4'31″ = 1.20655891055 rad

Height: ha = 19.61550452459
Height: hb = 14.01107466042
Height: hc = 14.01107466042

Median: ma = 19.61550452459
Median: mb = 14.92548115566
Median: mc = 14.92548115566

Inradius: r = 5.16218540121
Circumradius: R = 11.24113709597

Vertex coordinates: A[21; 0] B[0; 0] C[5.35771428571; 14.01107466042]
Centroid: CG[8.78657142857; 4.67702488681]
Coordinates of the circumscribed circle: U[10.5; 4.01547753427]
Coordinates of the inscribed circle: I[7.5; 5.16218540121]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.1550335145° = 138°9'1″ = 0.73304144426 rad
∠ B' = β' = 110.9254832428° = 110°55'29″ = 1.20655891055 rad
∠ C' = γ' = 110.9254832428° = 110°55'29″ = 1.20655891055 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 = 21 ; ; c = 21 ; ;

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

p = a+b+c = 15+21+21 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-15)(28.5-21)(28.5-21) } ; ; T = sqrt{ 21642.19 } = 147.11 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 147.11 }{ 15 } = 19.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 147.11 }{ 21 } = 14.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 147.11 }{ 21 } = 14.01 ; ;

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**2-21**2-21**2 }{ 2 * 21 * 21 } ) = 41° 50'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 69° 4'31" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-15**2-21**2 }{ 2 * 21 * 15 } ) = 69° 4'31" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 147.11 }{ 28.5 } = 5.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15 }{ 2 * sin 41° 50'59" } = 11.24 ; ;




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