Isosceles triangle calculator (b,h) - result

Please enter two properties of the isosceles triangle

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


You have entered side b and height hc.

Acute isosceles triangle.

Sides: a = 4.07770700264   b = 4.07770700264   c = 4.5

Area: T = 7.65
Perimeter: p = 12.65441400528
Semiperimeter: s = 6.32770700264

Angle ∠ A = α = 56.50548153263° = 56°30'17″ = 0.98661950707 rad
Angle ∠ B = β = 56.50548153263° = 56°30'17″ = 0.98661950707 rad
Angle ∠ C = γ = 66.99903693475° = 66°59'25″ = 1.16992025122 rad

Height: ha = 3.75326949258
Height: hb = 3.75326949258
Height: hc = 3.4

Median: ma = 3.77989714209
Median: mb = 3.77989714209
Median: mc = 3.4

Inradius: r = 1.20990904586
Circumradius: R = 2.44444852941

Vertex coordinates: A[4.5; 0] B[0; 0] C[2.25; 3.4]
Centroid: CG[2.25; 1.13333333333]
Coordinates of the circumscribed circle: U[2.25; 0.95655147059]
Coordinates of the inscribed circle: I[2.25; 1.20990904586]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 123.4955184674° = 123°29'43″ = 0.98661950707 rad
∠ B' = β' = 123.4955184674° = 123°29'43″ = 0.98661950707 rad
∠ C' = γ' = 113.0109630653° = 113°35″ = 1.16992025122 rad

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How did we calculate this triangle?

1. Input data entered: side b and height hc

b = 4.5 ; ; hc = 3.4 ; ;

2. From height h we calculate side a - Pythagorean theorem:

a**2 = h**2 + (c/2)**2 ; ; a = sqrt{ h**2 + (c/2)**2 } = sqrt{ 3.4**2 + (4.5/2)**2 } = 4.077 ; ;

3. From side a we calculate perimeter p:

p = 2a + c = 2 * 4.077 + 4.5 = 12.654 ; ;

4. From side a we calculate side b:

b = a = 4.077 ; ;
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 = 4.08 ; ; b = 4.08 ; ; c = 4.5 ; ;

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

p = a+b+c = 4.08+4.08+4.5 = 12.65 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 12.65 }{ 2 } = 6.33 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.33 * (6.33-4.08)(6.33-4.08)(6.33-4.5) } ; ; T = sqrt{ 58.52 } = 7.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.65 }{ 4.08 } = 3.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.65 }{ 4.08 } = 3.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.65 }{ 4.5 } = 3.4 ; ;

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{ 4.08**2+4.5**2-4.08**2 }{ 2 * 4.08 * 4.5 } ) = 56° 30'17" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 4.08**2+4.5**2-4.08**2 }{ 2 * 4.08 * 4.5 } ) = 56° 30'17" ; ; gamma = 180° - alpha - beta = 180° - 56° 30'17" - 56° 30'17" = 66° 59'25" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.65 }{ 6.33 } = 1.21 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 4.08 }{ 2 * sin 56° 30'17" } = 2.44 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.08**2+2 * 4.5**2 - 4.08**2 } }{ 2 } = 3.779 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.5**2+2 * 4.08**2 - 4.08**2 } }{ 2 } = 3.779 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.08**2+2 * 4.08**2 - 4.5**2 } }{ 2 } = 3.4 ; ;
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