# Equilateral Triangle

## Sum of Areas of Equilateral Triangles Inscribed in Circles

**Problem**

An equilateral triangle is inscribed within a circle whose diameter is 12 cm. In this triangle a circle is inscribed; and in this circle, another equilateral triangle is inscribed; and so on indefinitely. Find the sum of the areas of all the triangles.

## 09 Dimensions of smaller equilateral triangle inside the circle

**Problem**

From the figure shown, ABC and DEF are equilateral triangles. Point E is the midpoint of AC and points D and F are on the circle circumscribing ABC. If AB is 12 cm, find DE.

## 10 Area common to three squares inside the regular hexagon

**Problem**

Three squares are drawn so that each will contain a side of regular hexagon as shown in the given figure. If the hexagon has a perimeter of 60 in., compute the area of the region common to the three squares. The required area is the shaded region in the figure.

## 06 Circular arcs inside and tangent to an equilateral triangle

**Example 06**

The figure shown below is an equilateral triangle of sides 20 cm. Three arcs are drawn inside the triangle. Each arc has center at one vertex and tangent to the opposite side. Find the area of region enclosed by these arcs. The required area is shaded as shown in the figure below.

## 05 Three identical cirular arcs inside a circle

## The Regular Tetrahedron

Regular tetrahedron is one of the regular polyhedrons. It is a triangular pyramid whose faces are all equilateral triangles.