Question

The diameter of a circle equals the diagonal of a square whose side length is 4 $$\begin{array}{cc}\text { Quantity } \mathbf{A} & \text { Quantity } \mathbf{B} \\ \text {The circumference of the circle } & \text {\(\sqrt{101}\)} \end{array}$$ A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

Short answer

A. Quantity A is greater.

Step-by-step solution

Calculate the diagonal of the square

The side length of the square is given as 4. Use the Pythagorean Theorem to find the diagonal. The diagonal (d) of the square can be found using the formula: .

Calculate the circumference of the circle

The diameter of the circle is equal to the diagonal of the square, which is . The circumference (C) of the circle can be found using the formula: \[C = \pi \times d\].

Compare the results

Calculate the numerical values and then compare Quantity A and Quantity B. From Step 2, the circumference of the circle is \( \pi \times \sqrt{32} \). Calculate: \( \approx 5.6568 \pi \approx 17.78 \). Comparing \(17.78\) with \(10.05\) shows Quantity A is greater.

Key concepts

circle circumference calculation

To find the circumference of a circle, you need to know the diameter. The formula for circumference is: \[C = \pi \times d\]. Where \[C\] stands for circumference and \[d\] stands for diameter. This formula tells us how far we would travel if we walked around the circle once. If you only know the radius (r), you can find the diameter using the formula: \[d = 2 \times r\].
For example, if the diameter is 10, then the circumference will be \[C = \pi \times 10= 31.42\], assuming \[ \pi \] is approximately 3.14.
If the diameter is given by a more complex number like \[ \sqrt{32} \], then circumference becomes more interesting. Here, \[ C = \pi \times \sqrt{32} \].
This kind of problem is common in geometry, so make sure you're comfortable using \[ \pi \] and basic formulas.

diagonal of square

Finding the diagonal of a square is straightforward if you understand the Pythagorean Theorem. When you have a square, all sides are of equal length. Let's call the side length \[ s \]. The diagonal splits the square into two right triangles. According to the Pythagorean Theorem, you can find the diagonal (d) using the formula: \[d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s \sqrt{2} \].
For example, if each side of the square is 4, the diagonal would be: \[4 \sqrt{2} = \sqrt{32} \]. So, the diagonal of a square with a side length of 4 is \[ \sqrt{32} \].
The diagonal is longer than any side of the square. This concept recurs in various geometry problems, so practice is key.

Pythagorean Theorem in geometry

The Pythagorean Theorem is a vital part of geometry that helps in finding a missing side of a right triangle. The theorem states: \[a^2 + b^2 = c^2\] where \[ a \] and \[ b \] are the legs of the triangle, and \[ c \] is the hypotenuse. This theorem is particularly useful when dealing with squares, rectangles, and other quadrilaterals.
For instance, if you have a right triangle with legs of lengths 3 and 4, you can find the hypotenuse as follows: \[ c = \sqrt{3^2 + 4^2 } = \sqrt{9 + 16} = \sqrt{25} = 5 \].

comparison of quantities

Comparing quantities is an essential skill often tested in exams like the GRE. Here, you need to analyze given information and compute values where necessary. For example, if given two quantities to compare, you first calculate each quantity.
Let's say we need to compare the circumference of a circle (Quantity A) with \[ \sqrt{101} \] (Quantity B). Calculate Quantity A first: if the diameter is \[ \sqrt{32} \], then \[ C = \pi \times \sqrt{32} \approx 17.78 \], because \[ \pi \approx 3.14 \].
Next, evaluate Quantity B, which is approximately 10.05. By comparing 17.78 with 10.05, it's clear that Quantity A is greater.
This exercise teaches you to perform calculations under time constraints and strengthens your logical comparison skills.

comparison of quantities

Comparing quantities is an essential skill often tested in exams like the GRE. Here, you need to analyze given information and compute values where necessary. For example, if given two quantities to compare, you first calculate each quantity.
Let's say we need to compare the circumference of a circle (Quantity A) with \[ \sqrt{101} \] (Quantity B). Calculate Quantity A first: if the diameter is \[ \sqrt{32} \], then \[ C = \pi \times \sqrt{32} \approx 17.78 \], because \[ \pi \approx 3.14 \].
Next, evaluate Quantity B, which is approximately 10.05. By comparing 17.78 with 10.05, it's clear that Quantity A is greater.
This exercise teaches you to perform calculations under time constraints and strengthens your logical comparison skills.