Question

The length of side of an equilateral triangle is a. There is circle inscribed in a triangle. What is the area of a square inscribed in a circle ? (a) \(\left(\mathrm{a}^{2} / 3\right)\) (b) \(\left(\mathrm{a}^{2} / 6\right)\) (c) \(\left(\mathrm{a}^{2} / \sqrt{3}\right)\) (d) \(\left(a^{2} / \sqrt{2}\right)\)

Short answer

The area of the square inscribed in the circle is \(\left(\frac{a^2}{6}\right)\).

Step-by-step solution

Find the radius of the inscribed circle

In an equilateral triangle of side length a, the radius (r) of the inscribed circle can be calculated using the formula: \(r = \frac{a}{2\sqrt{3}}\) This formula is derived from the fact that the centroid, circumcenter, and incenter of an equilateral triangle coincide, and the radius can be obtained by the ratio of the side length and the altitude.

Find the diameter of the inscribed circle

Since the diameter (d) is twice the radius (r) of the circle, we get: \(d = 2r = \frac{a}{\sqrt{3}}\)

Find the side length of the inscribed square

For a square inscribed in a circle, the diameter of the circle is equal to the side length of the square multiplied by the square root of 2. Therefore, we can find the side length of the square by dividing the diameter by the square root of 2: \(s = \frac{d}{\sqrt{2}} = \frac{a}{\sqrt{3}\cdot\sqrt{2}}\)

Calculate the area of the square

Now that we have the side length (s) of the square, we can calculate its area using the formula: Area of square = \(s^2\) Substituting the value of s, we get: Area of square = \(\left(\frac{a}{\sqrt{3}\cdot\sqrt{2}}\right)^2 = \frac{a^2}{2\cdot 3} = \frac{a^2}{6}\) So, the correct answer is: (b) \(\left(\frac{a^2}{6}\right)\)

Key concepts

Radius of Inscribed Circle

Understanding the radius of an inscribed circle, particularly within an equilateral triangle, is a building block of geometry. The radius is the straight line from the center of the circle to the perimeter. In an equilateral triangle, this is especially interesting because the point from which the radius is drawn, known as the incenter, is the same as the centroid and circumcenter due to the triangle's symmetry.

In simpler terms, the center of the circle is the meeting point of the triangle's angle bisectors. To compute the radius (r) of the inscribed circle within an equilateral triangle with side length a, we use the formula: \(r = \frac{a}{2\text{√}3}\). This stems from mutiple geometric principles, but chiefly because the height of an equilateral triangle divides it into two 30-60-90 right triangles, each having height as the longer leg which can be linked to the radius calculations through proportions inherent in this special right triangle.

Area of Inscribed Square

The area of a square inscribed in a circle is an elegant measure of how two different shapes can relate to each other geometrically. The square inscribes within the circle in such a way that each of its vertices touches the circumference of the circle. To find the area of such a square, you need to understand that its diagonal is equal to the diameter of the circle.

Thus, if the diameter of the circle is known, the side of the square (\(s\)) can be found by the formula \( s = \frac{d}{\text{√}2} \). Once the side length is determined, the area is simply \( s^2 \). If the circle is inscribed in an equilateral triangle, we find the diameter from the triangle's side using geometric relationships, which then allows us to determine the area of the square.

Geometric Relationships in Polygons

Geometric relationships within polygons unfold the intricate symmetries and properties connecting different shapes. An understanding of how these shapes interact is fundamental. For instance, an inscribed figure, like a square or a circle, maintains constant distances to certain points of the polygons which enfold them.

The concept of inscribed and circumscribed figures revolves around these mutual geometric relations. In polygons, like triangles and squares, angles, sides, and other components such as altitude, diagonals, and radius display special relationships, like those derived in our main exercise. Knowing these relationships is powerful—they provide shortcuts and insights into solving various geometric problems.

Equilateral Triangle Properties

An equilateral triangle is a polygon with three equal sides and three equal angles, each measuring 60 degrees. The properties of an equilateral triangle are fascinating and valuable in the study of geometry. Due to this equal-sided symmetry, the incenter, circumcenter, centroid, and orthocenter all coincide at a single point within the triangle.

This inherent symmetry allows for easier calculations of area, height, and other related measures by providing consistent geometric relationships. The radius of an inscribed circle and the side length of the triangle are always in the same ratio, making it a perfect shape for foundational geometric explorations and inscribed figure exercises.