Question

What is the radius (in cm) of a circle whose circumference happens to equal its area (in cm2 )? (A) 2 cm (B) 3 cm (C) 5 cm (D) 10 cm

Short answer

The radius of a circle, whose circumference is equal to its area, is \(2 \: cm\) (Option A).

Step-by-step solution

Write down the circumference and area formulas for a circle

The circumference (C) of a circle is given by the formula: C = 2πr The area (A) of a circle is given by the formula: A = πr^2

Set the circumference equal to the area

Since the circumference equals the area, we can write: 2πr = πr^2

Solve for the radius (r)

To solve for the radius, follow these steps: 1. Divide both sides of the equation by π: 2r = r^2 2. Move all terms to one side of the equation: r^2 - 2r = 0 3. Factor the equation: r(r - 2) = 0 At this point, we have two possible solutions for r: r = 0 or r - 2 = 0 ⟹ r = 2 Since a circle with a radius of 0 is not truly a circle, we can disregard this solution.

Choose the correct answer

The radius of a circle, whose circumference is equal to its area, is 2 cm (Option A).

Key concepts

GED Math Practice

GED math practice typically involves reinforcing fundamental concepts and acquiring the skills necessary to solve a variety of mathematical problems, including geometry. One of the essential skills you will master is working with the properties of circles, such as understanding and calculating their circumference and area.

For students preparing for the GED, problems like the one about a circle's circumference being equal to its area test both algebra and geometry knowledge. To tackle these challenges effectively, it's important to become comfortable with the formulas for the circumference (\( C = 2\text{\textpi}r \) ) and area (\( A = \text{\textpi}r^2 \) ) of a circle. Practice solving for a missing variable using the interplay between these two formulas. Remember, cross-disciplinary exercises enhance problem-solving skills - a key component of the GED math section.

Geometry Problems

In geometry problems, particularly those relating to circles, you’re often asked to calculate key parameters such as radius, diameter, area, and circumference. Understanding the relationship between them is vital. The exercise provided is a classic example of how these relationships come to play - where the area and circumference are set equal.

When working on geometry problems involving circles, always start by noting what information is given and what needs to be found. Sketching the circle and labeling known and unknown values can provide visual clues to help solve the problem. In teaching geometry, it's essential to emphasize the process of creating equations based on the relationships between the circle's elements - as done in steps 2 and 3 of the solution. These steps are vital in making connections between algebraic expressions and geometrical figures.

Solving Circle Equations

Solving circle equations requires a blend of algebraic manipulation and understanding geometric principles. The process often involves setting up an equation based on the known formulas for the circumference and the area of a circle and then solving for the unknown variable.

In the example provided, the equation \( 2\text{\textpi}r = \text{\textpi}r^2 \) was constructed by setting the circumference equal to the area. To solve for the radius (\( r \) ), the equation was simplified by dividing both sides by \text{\textpi}, and then factoring to find the possible values of \( r \) . This step-by-step approach is essential. It's helpful to remind students to look out for unnecessary solutions (like a radius of 0, which doesn’t make sense for a circle) and cross-check with the physical parameters of the shapes in question.