Geometry Equations

Sample Prompt

Create a list of the 25 most commonly used
equations in a high school geometry course.

For each equation, return the following:

1. The name of the equation in markdown level 4.
2. Within the body, return the equation in LaTeX format
3. Surround the equation with dollar signs
4. For each variable in the equation, add a "where:" that defines each variable
5. Put blank lines around the list of variables
6. Wrap the variables names in dollar signs

Examples:

#### Slope and Intercept
$y = m * x + b$

where:

- $y$ is the vertical axis
- $x$ is the horizontal axis
- $m$ is the slope
- $b$ is the intercept

#### Area of a Circle
$A = 2\pi r^2$
where:

- $A$ is the area of the circle
- $r$ is the radius

Sample Response from ChatGPT

Note

We had to do extensive filtering on this because the Markdown Paste does not work correctly on LaTeX equations in VSCode. See the src/latex-cleanup code for details on the Python filter we had ChatGPT create to clean up the formatting.

1. Area of a Triangle

$A = \frac{1}{2} b h$

where:

A is the area of the triangle
b is the base of the triangle
h is the height of the triangle

2. Pythagorean Theorem

$c^{2} = a^{2} + b^{2}$

where:

$c$ is the hypotenuse of the right triangle
$a$ and $b$ are the other two sides

3. Area of a Circle

$A = 2 π r^{2}$

where:

$A$ is the area of the circle
$r$ is the radius

4. Circumference of a Circle

$C = 2 π r$

where:

$C$ is the circumference
$r$ is the radius

5. Distance Formula

$d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$

where:

$d$ is the distance between two points
$(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are the coordinates of the two points

6. Midpoint Formula

$M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

where:

M is the midpoint of the segment
$(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are the endpoints

7. Slope Formula

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

where:

$m$ is the slope
$(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are the points

8. Area of a Rectangle

$A = l \cdot w$ where:

$A$ is the area
$l$ is the length
$w$ is the width

9. Perimeter of a Rectangle

$P = 2 l + 2 w$

where:

$P$ is the perimeter
$l$ is the length
$w$ is the width

10. Area of a Parallelogram

$A = b h$

where:

$A$ is the area
$b$ is the base
$h$ is the height

11. Area of a Trapezoid

$A = \frac{1}{2} (b_{1} + b_{2}) h$

where:

$A$ is the area
$b_{1}$ and $b_{2}$ are the bases
$h$ is the height

12. Volume of a Rectangular Prism

$V = l \cdot w \cdot h$

where:

$V$ is the volume
$l$ is the length
$w$ is the width
$h$ is the height

13. Surface Area of a Rectangular Prism

$S A = 2 l w + 2 l h + 2 w h$ where:

$S A$ is the surface area
$l$ is the length
$w$ is the width
$h$ is the height

14. Volume of a Sphere

$V = \frac{4}{3} π r^{3}$

where:

$V$ is the volume
$r$ is the radius

15. Surface Area of a Sphere

$S A = 4 π r^{2}$ where:

$S A$ is the surface area
$r$ is the radius

16. Volume of a Cylinder

$V = π r^{2} h$

where:

$V$ is the volume
$r$ is the radius
$h$ is the height

17. Surface Area of a Cylinder

$S A = 2 π r^{2} + 2 π r h$

where:

$S A$ is the surface area
$r$ is the radius
$h$ is the height

18. Volume of a Cone

$V = \frac{1}{3} π r^{2} h$

where:

$V$ is the volume
$r$ is the radius
$h$ is the height

19. Surface Area of a Cone

$S A = π r (r + l)$

where:

$S A$ is the surface area
$r$ is the radius
$l$ is the slant height

20. Area of a Regular Polygon

$A = \frac{1}{2} a P$

where:

$A$ is the area
$a$ is the apothem
$P$ is the perimeter

21. Sum of Interior Angles of a Polygon

$S = 180 (n - 2)$

where:

$S$ is the sum of the interior angles
$n$ is the number of sides

22. Measure of Each Interior Angle of a Regular Polygon

$A n g l e = \frac{180 (n - 2)}{n}$

where:

$A n g l e$ is the measure of each interior angle
$n$ is the number of sides

23. Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

where:

$a, b, c$ are the sides of the triangle
$A, B, C$ are the angles opposite those sides

24. Law of Cosines

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$

where:

$c$ is the side opposite angle C
$a$ and b are the other sides
$C$ is the included angle

25. Equation of a Circle

$(x - h) 2 + (y - k) 2 = r^{2}$

where:

$(h, k)$ is the center of the circle
$r$ is the radius