Geometry Equations
Sample Prompt
Prompt
Create a list of the 25 most commonly used equations in a high school geometry course.
For each equation, return the following:
- The name of the equation in markdown level 4.
- Within the body, return the equation in LaTeX format
- Surround the equation with dollar signs
- For each variable in the equation, add a "where:" that defines each variable
- Put blank lines around the list of variables
- 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
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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
\(A = 2\pi r^2\)
where:
- \(A\) is the area of the circle
- \(r\) is the radius
4. Circumference of a Circle
\(C = 2 \pi r\)
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
\(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
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
\(P = 2l + 2w\)
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
\(SA = 2lw + 2lh + 2wh\) where:
- \(SA\) is the surface area
- \(l\) is the length
- \(w\) is the width
- \(h\) is the height
14. Volume of a Sphere
\(V = \frac{4}{3} \pi r^3\)
where:
- \(V\) is the volume
- \(r\) is the radius
15. Surface Area of a Sphere
\(SA = 4 \pi r^2\) where:
- \(SA\) is the surface area
- \(r\) is the radius
16. Volume of a Cylinder
\(V = \pi r^2 h\)
where:
- \(V\) is the volume
- \(r\) is the radius
- \(h\) is the height
17. Surface Area of a Cylinder
\(SA = 2\pi r^2 + 2\pi r h\)
where:
- \(SA\) is the surface area
- \(r\) is the radius
- \(h\) is the height
18. Volume of a Cone
\(V = \frac{1}{3} \pi r^2 h\)
where:
- \(V\) is the volume
- \(r\) is the radius
- \(h\) is the height
19. Surface Area of a Cone
\(SA = \pi r (r + l)\)
where:
- \(SA\) 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
\(Angle = \frac{180(n - 2)}{n}\)
where:
- \(Angle\) 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
\(c^2 = a^2 + b^2 - 2ab\cos C\)
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