Volume of Triangle w/ xy-plane Vertices - Help Needed

  • Context: Undergrad 
  • Thread starter Thread starter gillyr2
  • Start date Start date
  • Tags Tags
    Volume
Click For Summary
SUMMARY

The discussion focuses on calculating the volume of a solid with a triangular base defined by the vertices (0,0), (1,0), and (0,1) in the xy-plane. The solid's cross sections, which are perpendicular to the y-axis, are equilateral triangles. The area of the triangular base is established as 1/2 bh, with the height of the equilateral triangles being derived from the relationship h = (sqrt(3)/4)(1-y), where x = 1 - y. The participants clarify the geometric properties of equilateral triangles to assist in solving the volume calculation.

PREREQUISITES
  • Understanding of basic geometry, specifically properties of equilateral triangles.
  • Knowledge of calculus, particularly integration techniques for volume calculation.
  • Familiarity with the concept of cross sections in three-dimensional solids.
  • Ability to manipulate algebraic expressions involving variables.
NEXT STEPS
  • Study the method of calculating volumes of solids with known cross-sectional areas.
  • Learn about integration techniques for solids of revolution.
  • Explore the properties of equilateral triangles and their geometric relationships.
  • Research the application of the Fundamental Theorem of Calculus in volume problems.
USEFUL FOR

Students in calculus courses, educators teaching geometry and volume calculations, and anyone involved in mathematical modeling of three-dimensional shapes.

gillyr2
Messages
45
Reaction score
0
Question: Find the volume of the solid whose base is the triangular region of the xy-plane with vertices (0,0),(1,0),(0,1) and whose cross sections perpendicular to the y-axis are equilateral triangles.

I have the problem set up. just don't know how to get the cross sections of the triangles. i know the area is 1/2 bh i thought maybe similar triangles where the height would be sqrt(3)/4x and x=1-y. am i close? help
 
Physics news on Phys.org
gillyr2 said:
i know the area is 1/2 bh i thought maybe similar triangles where the height would be sqrt(3)/4x and x=1-y. am i close? help

Hi gillyr2! :smile:

yeah … close … :wink:

though I don't understand what you mean by "similar triangles" …

these are equilateral (3 x 60º) triangles of side (as you say) 1 - x …

so h = … ? :smile:
 

Similar threads

MHB What are the y limits for finding the volume of a solid with given parameters?
  • · Replies 5 ·
Replies
5
Views
2K
MHB What is the Volume of a Solid with Squares as Cross Sections?
  • · Replies 2 ·
Replies
2
Views
2K
MHB College-_-'s question at Yahoo Answers regarding a volume by slicing
  • · Replies 3 ·
Replies
3
Views
19K
MHB Volume of Solid with Circular Base & Equilateral Triangles
  • · Replies 4 ·
Replies
4
Views
8K
MHB Nick's question at Yahoo Answers regarding a volume by slicing
  • · Replies 1 ·
Replies
1
Views
2K
MHB Airshow4444's question at Yahoo Answers regarding volumes by slicing
  • · Replies 1 ·
Replies
1
Views
3K
MHB Changing variables for defining a region
  • · Replies 2 ·
Replies
2
Views
2K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Proof that lattice points can't form an equilateral triangle
  • · Replies 5 ·
Replies
5
Views
5K
MHB Nick's question at Yahoo Answers regarding a volume by slicing
  • · Replies 1 ·
Replies
1
Views
2K