This circle is centered at the origin, and the length of its radius is 3. What is the circle's equation?
A. $x^2+y^2=9$
B. $x^3+y^3=27$
C. $x^2+y^2=3$
D. $x+y=3$

The problem provides a circle centered at the origin with a radius of 3.
The general equation of a circle is ( x − h ) 2 + ( y − k ) 2 = r 2 , where (h, k) is the center and r is the radius.
Substituting the given values (h = 0, k = 0, r = 3) into the general equation yields x 2 + y 2 = 3 2 .
Simplifying, the equation of the circle is x 2 + y 2 = 9 ​ .

Explanation

Problem Analysis We are given a circle centered at the origin (0, 0) with a radius of 3. Our goal is to find the equation of this circle.

Recall the general equation of a circle The general equation of a circle with center (h, k) and radius r is given by: ( x − h ) 2 + ( y − k ) 2 = r 2

Substitute the given values In our case, the center is at the origin, so h = 0 and k = 0. The radius is given as r = 3. Substituting these values into the general equation, we get: ( x − 0 ) 2 + ( y − 0 ) 2 = 3 2

Simplify the equation Simplifying the equation, we have: x 2 + y 2 = 9

State the final answer Therefore, the equation of the circle centered at the origin with a radius of 3 is x 2 + y 2 = 9 .


Examples
Circles are fundamental in many real-world applications. For example, in architecture, arches and domes often utilize circular shapes for their structural integrity and aesthetic appeal. In engineering, gears and wheels are based on circular designs to ensure smooth and efficient motion. Understanding the equation of a circle allows engineers and architects to precisely design and construct these structures, ensuring they meet specific requirements and perform optimally. For instance, knowing the equation of a circle helps in calculating the area and circumference, which are crucial for material estimation and structural analysis.

Answered by GinnyAnswer | 2025-07-05