7 21 21 triangle

Acute isosceles triangle.

Sides: a = 7   b = 21   c = 21

Area: T = 72.4721977343
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 19.18881364537° = 19°11'17″ = 0.33548961584 rad
Angle ∠ B = β = 80.40659317731° = 80°24'21″ = 1.40333482476 rad
Angle ∠ C = γ = 80.40659317731° = 80°24'21″ = 1.40333482476 rad

Height: ha = 20.70662792408
Height: hb = 6.90220930803
Height: hc = 6.90220930803

Median: ma = 20.70662792408
Median: mb = 11.60881867662
Median: mc = 11.60881867662

Inradius: r = 2.95880398915
Circumradius: R = 10.64989436096

Vertex coordinates: A[21; 0] B[0; 0] C[1.16766666667; 6.90220930803]
Centroid: CG[7.38988888889; 2.30106976934]
Coordinates of the circumscribed circle: U[10.5; 1.77548239349]
Coordinates of the inscribed circle: I[3.5; 2.95880398915]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.8121863546° = 160°48'43″ = 0.33548961584 rad
∠ B' = β' = 99.59440682269° = 99°35'39″ = 1.40333482476 rad
∠ C' = γ' = 99.59440682269° = 99°35'39″ = 1.40333482476 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 = 7 ; ; b = 21 ; ; c = 21 ; ;

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

p = a+b+c = 7+21+21 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-7)(24.5-21)(24.5-21) } ; ; T = sqrt{ 5252.19 } = 72.47 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72.47 }{ 7 } = 20.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72.47 }{ 21 } = 6.9 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72.47 }{ 21 } = 6.9 ; ;

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{ 21**2+21**2-7**2 }{ 2 * 21 * 21 } ) = 19° 11'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7**2+21**2-21**2 }{ 2 * 7 * 21 } ) = 80° 24'21" ; ; gamma = 180° - alpha - beta = 180° - 19° 11'17" - 80° 24'21" = 80° 24'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72.47 }{ 24.5 } = 2.96 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7 }{ 2 * sin 19° 11'17" } = 10.65 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 21**2 - 7**2 } }{ 2 } = 20.706 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 7**2 - 21**2 } }{ 2 } = 11.608 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 7**2 - 21**2 } }{ 2 } = 11.608 ; ;
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