An equilateral triangle has two equal sides with the angles opposite to them equal. The relationship between the lateral side \( a \), the based \( b \) of the isosceles triangle, its area A, height h, inscribed and circumscribed radii r and R respectively are give by:
Area A of triangle = \( \dfrac{1}{2} b h = \dfrac{1}{2} a^2 \sin(\alpha) \)
b = \( 2 a \cos (\beta) \)
lenght of height h perpendicular to the base
h = \( a \cos (\alpha/2) \)
R = \( \dfrac{a}{2\sin(\beta)} = \dfrac{a^2}{\sqrt{4a^2b^2}} \)
r = \( \dfrac{b}{2} \sqrt{\dfrac{2ab}{2a+b}}\)
Problems with Solutions

What is the area of an isosceles triangle with base b of 8 cm and lateral a side 5 cm?

What is the base of an isosceles triangle with lateral side a = 5 cm and area 6 cm^{ 2}?

What is the lateral side of an isosceles triangle with area 20 unit^{ 2} and base 10 units?

What is the lateral side of an isosceles triangle such that its height h ( perpendicular to its base b) is 4 cm shorter than its base b and its area is 30 cm^{ 2}?

ABC and BCD are isosceles triangles. Find the size of angle BDE.

ABC and CDE are isosceles triangles. Find the size of angle CED.

Find the area of the circle inscribed to an isosceles triangle of base 10 units and lateral side 12 units.

Find the ratio of the radii of the circumscribed and inscribed circles to an isosceles triangle of base b units and lateral side a units such that a = 2 b.

Find the lateral side and base of an isosceles triangle whose height ( perpendicular to the base ) is 16 cm and the radius of its circumscribed circle is 9 cm.

What is the area of an isosceles triangle of lateral side 2 units that is similar to another isosceles triangle of lateral side 10 units and base 12 units?