What is the area of a quadrant? (1) The arc is \(4.56 \mathrm{~cm}\) (2) Its radius is \(4 \mathrm{~cm}\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

A quadrant is essentially one-quarter of a circle. To calculate the area of a quadrant, you first need to understand the formula for the area of a full circle: \(\text{Area} = \pi r^2\). For a quadrant, you simply take one-fourth of this area, so the formula becomes: \[\text{Quadrant Area} = \frac{1}{4} \pi r^2\]. If the radius \(r\) is given, you can easily substitute it into this formula.
To illustrate this, consider a circle with a radius of 4 cm. The total area is \[ \pi (4)^2 \text{cm}^2 \]. Therefore, the area of the quadrant is: \[\text{Quadrant Area} = \frac{1}{4} \pi (4)^2 \].
If you have the arc length of the quadrant instead, as given in Statement 1, you need to convert that information to find the radius. The arc length formula is: \[\text{Arc Length} = \frac{1}{4} (2 \pi r) = \frac{\pi r}{2}\ \text{cm} \]. Solving for \(r\), you use the equation \(\frac{\pi r}{2} = 4.56 \text{cm}\) to find \((r)\). However, if this radius doesn’t match other given information, it makes the statement alone insufficient to calculate the quadrant area.

Understanding how to calculate the circumference and area of a circle is essential for tackling problems involving quadrants. The circumference formula is: \(C = 2 \pi r\). This tells you the total outer length of the circle. Using the circumference, you can derive the radius if needed.
Consider Statement 1 from the exercise, which provides the arc length of the quadrant as 4.56 cm. Since the arc length is one-quarter of the full circumference, set up the equation: \[\frac{1}{4} 2 \pi r = 4.56\]
Solving for \(r\), you get: \[\frac{\pi r}{2} = 4.56\]
\text{or} \ \[ r = \frac{9.12}{\pi} \approx 2.9 \text{cm} \]. This allows you to solve for the radius, which can then be used in the area equation: \( \text{Area} = \pi r^2 \).
Finally, the area of a full circle with this radius can be divided by four to get the quadrant area. Mastering these two formulas will greatly aid in solving a wide range of GMAT geometry problems.

When approaching GMAT geometry problems, it helps to follow a structured problem-solving process. Here are some tips:
1. **Read the Problem Carefully** - Understand what is being asked. Identify key information and rephrase the problem in your own words if needed.
2. **Identify Formulas and Relationships** - Determine which formulas or theorems apply to the problem. In the case of a quadrant, formulas such as \( \text{Area} = \pi r^2 \) or \(\text{Arc Length} = \frac{1}{4} 2 \pi r\) are relevant.
3. **Evaluate Each Statement Independently** - For questions with multiple parts, assess each piece of information separately to see if it’s sufficient. In our example, we evaluated Statements 1 and 2 to determine whether they independently provided enough details to calculate the area of the quadrant.
4. **Combine Information if Needed** - If individual statements are insufficient, consider whether combining them offers a solution. Always check the interplay between different pieces of information.
5. **Perform Accurate Calculations** - Simplify fractions, and carefully compute values. Double-check your steps to ensure no mistakes in simple arithmetic.
Following these steps will help you systematically work through GMAT geometry problems and improve your accuracy in reaching solutions.