--- title: Circle TARGET DECK: Obsidian::STEM FILE TAGS: geometry::circle tags: - circle - geometry --- ## Overview A **circle** is a shape consisting of all points in a plane at a given distance from a given point, i.e. the **center**. In this way, a circle is characterized by its center and **radius**, the distance from the center to any point of the circle. The perimeter of a circle is known as its **circumference**. On the [[cartesian|Cartesian coordinate system]], the equation of a circle with radius $r$ and center $\langle h, k \rangle$ is $$(x - h)^2 + (y - k)^2 = r^2.$$ Such a circle has circumference $2\pi r$ and [[area]] $\pi r^2$. A continuous segment along the circumference of a circle is called an **arc**. %%ANKI Basic A circle is characterized by what two properties? Back: Its center and its radius. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic In plain English, describe what a circle is. Back: A set of points at a given distance from some given point. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic The perimeter of a circle is known as what? Back: Its circumference. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic What is the Cartesian equation of a circle with radius $r$ and center $\langle h, k \rangle$? Back: $(x - h)^2 + (y - k)^2 = r^2$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Given indeterminates $x$ and $y$, the following equation describes what shape? $$(x - h)^2 + (y - k)^2 = r^2$$ Back: A circle with radius $r$ and center $\langle h, k \rangle$. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Consider a circle with radius $r$. What does its circumference evaluate to? Back: $2 \pi r$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Consider a circle with radius $r$. What is its area? Back: $\pi r^2$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Consider a circle with diameter $d$. What does its circumference evaluate to? Back: $\pi d$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Consider a circle with diameter $d$. What its area? Back: $\pi \left(\frac{d}{2}\right)^2$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Consider a circle with radius $r$. What does the following evaluate to? $$2 \pi r$$ Back: Its circumference. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Consider a circle with radius $r$. What does the following evaluate to? $$\pi r^2$$ Back: Its area. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic What is the Cartesian equation of the following shape? ![[unit-circle.png]] Back: $x^2 + y^2 = 1$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic What is the Cartesian equation of the following shape? ![[circle-right.png]] Back: $(x - 2)^2 + y^2 = 4$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic What is the Cartesian equation of the following shape? ![[circle-left-down.png]] Back: $(x + 1)^2 + (y + 1)^2 = 4$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic What is the Cartesian equation of the following shape? ![[circle-left-up.png]] Back: $(x + 1)^2 + (y - 1)^2 = 4$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Rewrite equation $x^2 + y^2 = 1$ shifted left by $a > 0$. Back: $(x + a)^2 + y^2 = 1$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Rewrite equation $x^2 + y^2 = 1$ shifted up by $b > 0$. Back: $x^2 + (y - b)^2 = 1$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Rewrite equation $x^2 + y^2 = 1$ shifted right by $a > 0$. Back: $(x - a)^2 + y^2 = 1$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic Rewrite equation $x^2 + y^2 = 1$ shifted down by $b > 0$. Back: $x^2 + (y + b)^2 = 1$ Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic In what direction(s) is the unit circle shifted in the following? $$(x - 3)^2 + (y - 3)^2 = 1$$ Back: Right by $3$ and up by $3$. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic In what direction(s) is the unit circle shifted in the following? $$(x + 3)^2 + (y - 3)^2 = 1$$ Back: Left by $3$ and up by $3$. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic In what direction(s) is the unit circle shifted in the following? $$(x - 3)^2 + (y + 3)^2 = 1$$ Back: Right by $3$ and down by $3$. Reference: “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). END%% %%ANKI Basic What term is used to describe a continuous portion of a circle? Back: An arc. Reference: Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024. END%% ## Bibliography * “Circle,” in _Wikipedia_, January 8, 2025, [https://en.wikipedia.org/w/index.php?title=Circle](https://en.wikipedia.org/w/index.php?title=Circle&oldid=1268270102). * Ted Sundstrom and Steven Schlicker, _Trigonometry_, 2024.