10 15 15 triangle

Acute isosceles triangle.

Sides: a = 10   b = 15   c = 15

Area: T = 70.71106781187
Perimeter: p = 40
Semiperimeter: s = 20

Angle ∠ A = α = 38.9422441269° = 38°56'33″ = 0.68796738189 rad
Angle ∠ B = β = 70.52987793655° = 70°31'44″ = 1.23109594173 rad
Angle ∠ C = γ = 70.52987793655° = 70°31'44″ = 1.23109594173 rad

Height: ha = 14.14221356237
Height: hb = 9.42880904158
Height: hc = 9.42880904158

Median: ma = 14.14221356237
Median: mb = 10.3087764064
Median: mc = 10.3087764064

Inradius: r = 3.53655339059
Circumradius: R = 7.95549512883

Vertex coordinates: A[15; 0] B[0; 0] C[3.33333333333; 9.42880904158]
Centroid: CG[6.11111111111; 3.14326968053]
Coordinates of the circumscribed circle: U[7.5; 2.65216504294]
Coordinates of the inscribed circle: I[5; 3.53655339059]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.0587558731° = 141°3'27″ = 0.68796738189 rad
∠ B' = β' = 109.4711220634° = 109°28'16″ = 1.23109594173 rad
∠ C' = γ' = 109.4711220634° = 109°28'16″ = 1.23109594173 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 = 10 ; ; b = 15 ; ; c = 15 ; ;

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

p = a+b+c = 10+15+15 = 40 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40 }{ 2 } = 20 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20 * (20-10)(20-15)(20-15) } ; ; T = sqrt{ 5000 } = 70.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 70.71 }{ 10 } = 14.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 70.71 }{ 15 } = 9.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 70.71 }{ 15 } = 9.43 ; ;

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{ 15**2+15**2-10**2 }{ 2 * 15 * 15 } ) = 38° 56'33" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+15**2-15**2 }{ 2 * 10 * 15 } ) = 70° 31'44" ; ; gamma = 180° - alpha - beta = 180° - 38° 56'33" - 70° 31'44" = 70° 31'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 70.71 }{ 20 } = 3.54 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 38° 56'33" } = 7.95 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 15**2 - 10**2 } }{ 2 } = 14.142 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 10**2 - 15**2 } }{ 2 } = 10.308 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 15**2+2 * 10**2 - 15**2 } }{ 2 } = 10.308 ; ;
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