209 lines
8.2 KiB
Markdown
209 lines
8.2 KiB
Markdown
---
|
|
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).
|
|
<!--ID: 1737167693438-->
|
|
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).
|
|
<!--ID: 1737167693441-->
|
|
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).
|
|
<!--ID: 1737167693445-->
|
|
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).
|
|
<!--ID: 1737167693448-->
|
|
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).
|
|
<!--ID: 1737167693451-->
|
|
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).
|
|
<!--ID: 1737167693455-->
|
|
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).
|
|
<!--ID: 1737167693458-->
|
|
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).
|
|
<!--ID: 1737167693462-->
|
|
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).
|
|
<!--ID: 1737167693465-->
|
|
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).
|
|
<!--ID: 1737167693468-->
|
|
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).
|
|
<!--ID: 1737167693472-->
|
|
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).
|
|
<!--ID: 1737167693476-->
|
|
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).
|
|
<!--ID: 1737167693481-->
|
|
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).
|
|
<!--ID: 1737167693485-->
|
|
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).
|
|
<!--ID: 1737167904156-->
|
|
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).
|
|
<!--ID: 1737168379559-->
|
|
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).
|
|
<!--ID: 1737168379562-->
|
|
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).
|
|
<!--ID: 1737168379565-->
|
|
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).
|
|
<!--ID: 1737168379568-->
|
|
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).
|
|
<!--ID: 1737168379571-->
|
|
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).
|
|
<!--ID: 1737168379574-->
|
|
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).
|
|
<!--ID: 1737168379578-->
|
|
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.
|
|
<!--ID: 1737347029852-->
|
|
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. |