21 25 25 triangle

Acute isosceles triangle.

Sides: a = 21   b = 25   c = 25

Area: T = 238.2255077395
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 49.66991749794° = 49°40'9″ = 0.86768906401 rad
Angle ∠ B = β = 65.16554125103° = 65°9'55″ = 1.13773510067 rad
Angle ∠ C = γ = 65.16554125103° = 65°9'55″ = 1.13773510067 rad

Height: ha = 22.68881026091
Height: hb = 19.05880061916
Height: hc = 19.05880061916

Median: ma = 22.68881026091
Median: mb = 19.41100489438
Median: mc = 19.41100489438

Inradius: r = 6.71105655604
Circumradius: R = 13.7743738835

Vertex coordinates: A[25; 0] B[0; 0] C[8.82; 19.05880061916]
Centroid: CG[11.27333333333; 6.35326687305]
Coordinates of the circumscribed circle: U[12.5; 5.78549703107]
Coordinates of the inscribed circle: I[10.5; 6.71105655604]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.3310825021° = 130°19'51″ = 0.86768906401 rad
∠ B' = β' = 114.835458749° = 114°50'5″ = 1.13773510067 rad
∠ C' = γ' = 114.835458749° = 114°50'5″ = 1.13773510067 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 = 25 ; ; c = 25 ; ;

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

p = a+b+c = 21+25+25 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-21)(35.5-25)(35.5-25) } ; ; T = sqrt{ 56751.19 } = 238.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 238.23 }{ 21 } = 22.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 238.23 }{ 25 } = 19.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 238.23 }{ 25 } = 19.06 ; ;

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{ 25**2+25**2-21**2 }{ 2 * 25 * 25 } ) = 49° 40'9" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 21**2+25**2-25**2 }{ 2 * 21 * 25 } ) = 65° 9'55" ; ; gamma = 180° - alpha - beta = 180° - 49° 40'9" - 65° 9'55" = 65° 9'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 238.23 }{ 35.5 } = 6.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 21 }{ 2 * sin 49° 40'9" } = 13.77 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 25**2 - 21**2 } }{ 2 } = 22.688 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 21**2 - 25**2 } }{ 2 } = 19.41 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 21**2 - 25**2 } }{ 2 } = 19.41 ; ;
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