Question

What is the ratio of the area of circumcircle of equilateral triangle to the area of square with the same side length as the equilateral triangle? (a) \(\pi: 3\) (b) \(\pi: \sqrt{3}\) (c) \(\sqrt{3}: 2\) (d) none of these

Short answer

Answer: (a) \(\frac{\pi}{3}\)

Step-by-step solution

Find the area of the circumcircle of the equilateral triangle

First, consider an equilateral triangle with side length 'a'. The radius of the circumcircle of an equilateral triangle can be found using the formula: \(R = \frac{a} {\sqrt{3}}\) Wheresa R is the radius of the circumcircle. Now, to find the area of the circumcircle, we will use the formula for the area of a circle: \(Area_{circle} = \pi R^2\) Substituting the value of \(R\) from the previous formula: \(Area_{circle} = \pi(\frac{a} {\sqrt{3}})^2\) \(Area_{circle} = \frac{\pi a^2}{3}\)

Find the area of the square with side length equal to the equilateral triangle

The area of a square can be calculated using the formula: \(Area_{square} = side^2\) Since the side length of the square is the same as the equilateral triangle, we have: \(Area_{square} = a^2\)

Calculate the ratio of the area of the circumcircle to the area of the square

Now, we can find the ratio between the area of the circumcircle and the area of the square: \(Ratio=\frac{Area_{circle}} {Area_{square}} = \frac{\frac{\pi a^2}{3}}{a^2}\) Upon simplifying the expression, we get: \(Ratio = \frac{\pi}{3}\) Since option (a) matches our result, the final answer is: (a) \(\pi : 3\)

Key concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length. This means each angle inside the triangle is also equal and measures 60 degrees. Equilateral triangles have a high degree of symmetry, which makes them a fascinating study in geometry.

• Each side of the triangle is denoted as 'a'.
• The heights and medians of an equilateral triangle are equal.
• They form natural patterns and are commonly seen in tiling and tessellation.

Due to its symmetry, many properties such as the radius of its circumcircle can be easily calculated. This makes them particularly interesting in geometric problems dealing with circles.

Circumcircle Area

The circumcircle of a triangle is a circle that passes through all three vertices of the triangle. For an equilateral triangle, finding this circle is straightforward due to its symmetry. The radius of the circumcircle for an equilateral triangle is given by the formula: \[ R = \frac{a}{\sqrt{3}} \]Here, 'R' is the radius and 'a' is the side length of the triangle.

• The area of the circumcircle is calculated using: \[ Area_{circle} = \pi R^2 \]
• By substituting the radius formula, the area becomes: \[ Area_{circle} = \frac{\pi a^2}{3} \]

This relationship shows how the side of an equilateral triangle affects the area of its circumcircle.

Square Area

A square is a regular quadrilateral, which means all four sides and all four angles are equal. If we have a square with the same side length 'a' as our equilateral triangle, calculating the area of the square is quite straightforward using the formula: \[ Area_{square} = a^2 \]
Squares are simple yet fundamental shapes in geometry, forming the basis of grids and many everyday design elements.

• All diagonals of a square are equal and intersect at right angles.
• Squares exhibit both rotational and reflectional symmetry.

The formula for the area is derived simply from multiplying the length of one side by itself.

Geometric Ratios

Geometric ratios involve comparing areas, lengths, or other quantities in geometry. In our problem, the focus is on the ratio of the circumcircle area of an equilateral triangle to the square area with the same side length. This ratio is expressed as: \[ Ratio = \frac{\text{Area of Circumcircle}}{\text{Area of Square}} = \frac{\frac{\pi a^2}{3}}{a^2} \]Simplifying this expression yields: \[ Ratio = \frac{\pi}{3} \]

• This illustrates the relationship between two geometric shapes based on a common measure (side 'a').
• It emphasizes the potency of equilateral triangle properties when combined with circular geometry.

Such exercises are crucial for understanding spatial relationships and comparing figures based on shared dimensions.