Isosceles triangle calculator (c)

Please enter two properties of the isosceles triangle

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


You have entered side c and angle γ.

Acute isosceles triangle.

Sides: a = 6.55500112243   b = 6.55500112243   c = 7.7

Area: T = 20.40114210162
Perimeter: p = 20.88000224486
Semiperimeter: s = 10.44000112243

Angle ∠ A = α = 54° = 0.94224777961 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 72° = 1.25766370614 rad

Height: ha = 6.22994308567
Height: hb = 6.22994308567
Height: hc = 5.29990703938

Median: ma = 6.35437911328
Median: mb = 6.35437911328
Median: mc = 5.29990703938

Inradius: r = 1.96216729806
Circumradius: R = 4.04881295633

Vertex coordinates: A[7.7; 0] B[0; 0] C[3.85; 5.29990703938]
Centroid: CG[3.85; 1.76663567979]
Coordinates of the circumscribed circle: U[3.85; 1.25109408305]
Coordinates of the inscribed circle: I[3.85; 1.96216729806]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126° = 0.94224777961 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 108° = 1.25766370614 rad

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How did we calculate this triangle?

1. Input data entered: side c and angle γ

c = 7.7 ; ; gamma = 72° ; ;

2. From side c we calculate side a - Pythagorean theorem:

a**2 = h**2 + (c/2)**2 ; ; a = sqrt{ h**2 + (c/2)**2 } = sqrt{ 5.299**2 + (7.7/2)**2 } = 6.55 ; ;

3. From side a and side c we calculate perimeter p:

p = 2a + c = 2 * 6.55 + 7.7 = 20.8 ; ;

4. From side a we calculate side b:

b = a = 6.55 ; ;
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 = 6.55 ; ; b = 6.55 ; ; c = 7.7 ; ;

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

p = a+b+c = 6.55+6.55+7.7 = 20.8 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.8 }{ 2 } = 10.4 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.4 * (10.4-6.55)(10.4-6.55)(10.4-7.7) } ; ; T = sqrt{ 416.22 } = 20.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20.4 }{ 6.55 } = 6.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20.4 }{ 6.55 } = 6.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20.4 }{ 7.7 } = 5.3 ; ;

9. 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{ 6.55**2+7.7**2-6.55**2 }{ 2 * 6.55 * 7.7 } ) = 54° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.55**2+7.7**2-6.55**2 }{ 2 * 6.55 * 7.7 } ) = 54° ; ; gamma = 180° - alpha - beta = 180° - 54° - 54° = 72° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20.4 }{ 10.4 } = 1.96 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 6.55 }{ 2 * sin 54° } = 4.05 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.55**2+2 * 7.7**2 - 6.55**2 } }{ 2 } = 6.354 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.7**2+2 * 6.55**2 - 6.55**2 } }{ 2 } = 6.354 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.55**2+2 * 6.55**2 - 7.7**2 } }{ 2 } = 5.299 ; ;
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