Pyramid question

A pyramid has a square base with sides of length 1 and has lateral faces that are equilateral triangles.

A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid.

Find volume of this cube.

#Geometry #HelpMe! #MathProblem #Math

Note by Piyushkumar Palan
7 years, 6 months ago

No vote yet
8 votes

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. list
  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Hi, check that a smaller square pyramid is being formed at the top having side same as that of cube (say ss),

Now, height of pyramid = s2\frac{s}{\sqrt{2}}

Now height from top to bottom = 12\frac{1}{\sqrt{2}},

Hence,

s+s2=12s + \frac{s}{\sqrt{2}} = \frac{1}{\sqrt{2}}

Solve to get volume = s3≈0.07106s^3 \approx 0.07106 cu. unit

jatin yadav - 7 years, 6 months ago

Log in to reply

Yes...

thats 52−7.. \boxed{5\sqrt2 - 7} .. thats the answer

Thanks ..

Piyushkumar Palan - 7 years, 6 months ago

let the horizontal side of the cube at the top equals x

let the vertical side of the cube equals y

let the height of the pyramid is h=1/2h = 1/\sqrt{2}

now x divides the lateral sides of the pyramid to two lengths ,let the top length equals k and the bottom length M

x=kx = k

since the base sides equals lateral sides of the pyramid is 1

Now y divides the lateral side of the cube to two lengths ,M and K

y/h=My/h = M

y=M/2y = M/\sqrt{2}

x=yx = y

thenk=M/2 k = M/\sqrt{2}

M+k=1M+k = 1

k+k∗2=1k + k*\sqrt{2}= 1

x/1=k/1x/1 = k/1

x=k/(k+k∗2)x = k/(k+k*\sqrt{2})

x=1/(1+2)x = 1/(1+\sqrt{2})

Mohamed Mohamed Dehairy - 7 years, 6 months ago

(Need Help)How to find the side of a regular tetrahedron which is inside a sphere of radius ' r ', Exactly fitting? Just as a triangle has circumcircle, say the Tetrahedron has a CIRCUMSPHERE...

Aditya Raut - 7 years, 6 months ago

Log in to reply

Aditya, i ve moved yours and more of mine questions on 3D geometry in New Discussion

here is the link https://brilliant.org/discussions/thread/3-d-geometry-questions/

Piyushkumar Palan - 7 years, 6 months ago

Log in to reply

Thank u very much sir :)

Aditya Raut - 7 years, 6 months ago

the answer is 0.0999 cubic units... let me know whether it is correct or not thanks in advance.

A Former Brilliant Member - 7 years, 6 months ago

Log in to reply

no sunny ... Jatin has got it right

Piyushkumar Palan - 7 years, 6 months ago
×

Problem Loading...

Note Loading...

Set Loading...