Isosceles triangle calculator (p)

Please enter two properties of the isosceles triangle

Use symbols: a, b, h, T, p, A, B, C, r, R


You have entered perimeter p and angle γ.

Right isosceles triangle.

Sides: a = 110.4210743493   b = 110.4210743493   c = 156.1598513015

Area: T = 6096.377029674
Perimeter: p = 377
Semiperimeter: s = 188.5

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 110.4210743493
Height: hb = 110.4210743493
Height: hc = 78.07992565073

Median: ma = 123.4544144288
Median: mb = 123.4544144288
Median: mc = 78.07992565073

Inradius: r = 32.34114869853
Circumradius: R = 78.07992565073

Vertex coordinates: A[156.1598513015; 0] B[0; 0] C[78.07992565073; 78.07992565073]
Centroid: CG[78.07992565073; 26.02664188358]
Coordinates of the circumscribed circle: U[78.07992565073; 0]
Coordinates of the inscribed circle: I[78.07992565073; 32.34114869853]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: angle γ and perimeter p

gamma = 90° ; ; p = 377 ; ;

2. From perimeter p we calculate side c:

p = 2a + c ; ; c = p - 2a = 377 - 2 * 110.421 = 156.159 ; ;

3. From we calculate side b:

b = a = 110.421 ; ;
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 = 110.42 ; ; b = 110.42 ; ; c = 156.16 ; ;

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

p = a+b+c = 110.42+110.42+156.16 = 377 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 377 }{ 2 } = 188.5 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 188.5 * (188.5-110.42)(188.5-110.42)(188.5-156.16) } ; ; T = sqrt{ 37165730.79 } = 6096.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6096.37 }{ 110.42 } = 110.42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6096.37 }{ 110.42 } = 110.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6096.37 }{ 156.16 } = 78.08 ; ;

8. 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{ 110.42**2+156.16**2-110.42**2 }{ 2 * 110.42 * 156.16 } ) = 45° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 110.42**2+156.16**2-110.42**2 }{ 2 * 110.42 * 156.16 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 45° - 45° = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6096.37 }{ 188.5 } = 32.34 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 110.42 }{ 2 * sin 45° } = 78.08 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 110.42**2+2 * 156.16**2 - 110.42**2 } }{ 2 } = 123.454 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 156.16**2+2 * 110.42**2 - 110.42**2 } }{ 2 } = 123.454 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 110.42**2+2 * 110.42**2 - 156.16**2 } }{ 2 } = 78.079 ; ;
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