Isosceles triangle calculator (b,h)

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 = 111.8033398875   b = 111.8033398875   c = 100

Area: T = 5000
Perimeter: p = 323.607679775
Semiperimeter: s = 161.8033398875

Angle ∠ A = α = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ B = β = 63.43549488229° = 63°26'6″ = 1.10771487178 rad
Angle ∠ C = γ = 53.13301023542° = 53°7'48″ = 0.9277295218 rad

Height: ha = 89.44327191
Height: hb = 89.44327191
Height: hc = 100

Median: ma = 90.13987818866
Median: mb = 90.13987818866
Median: mc = 100

Inradius: r = 30.90216994375
Circumradius: R = 62.5

Vertex coordinates: A[100; 0] B[0; 0] C[50; 100]
Centroid: CG[50; 33.33333333333]
Coordinates of the circumscribed circle: U[50; 37.5]
Coordinates of the inscribed circle: I[50; 30.90216994375]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ B' = β' = 116.5655051177° = 116°33'54″ = 1.10771487178 rad
∠ C' = γ' = 126.8769897646° = 126°52'12″ = 0.9277295218 rad

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How did we calculate this triangle?

1. Input data entered: side b and height hc

b = 100 ; ; hc = 100 ; ;

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{ 100**2 + (100/2)**2 } = 111.803 ; ;

3. From side a we calculate perimeter p:

p = 2a + c = 2 * 111.803 + 100 = 323.607 ; ;

4. From side a we calculate side b:

b = a = 111.803 ; ;
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 = 111.8 ; ; b = 111.8 ; ; c = 100 ; ;

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

p = a+b+c = 111.8+111.8+100 = 323.61 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 323.61 }{ 2 } = 161.8 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 161.8 * (161.8-111.8)(161.8-111.8)(161.8-100) } ; ; T = sqrt{ 25000000 } = 5000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5000 }{ 111.8 } = 89.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5000 }{ 111.8 } = 89.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5000 }{ 100 } = 100 ; ;

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{ 111.8**2+100**2-111.8**2 }{ 2 * 111.8 * 100 } ) = 63° 26'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 111.8**2+100**2-111.8**2 }{ 2 * 111.8 * 100 } ) = 63° 26'6" ; ;
 gamma = 180° - alpha - beta = 180° - 63° 26'6" - 63° 26'6" = 53° 7'48" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5000 }{ 161.8 } = 30.9 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 111.8 }{ 2 * sin 63° 26'6" } = 62.5 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.8**2+2 * 100**2 - 111.8**2 } }{ 2 } = 90.139 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 100**2+2 * 111.8**2 - 111.8**2 } }{ 2 } = 90.139 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 111.8**2+2 * 111.8**2 - 100**2 } }{ 2 } = 100 ; ;
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