Questions tagged [solid-geometry]

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. (Ref: http://en.m.wikipedia.org/wiki/Solid_geometry)

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Why is the number of ways to open up a cube $4! \times 2^4$?

A cube has $12$ edges. Cut seven of them and lay out the remaining ones on a table. It's known that the number of distinct connected meshes with non-overlapping faces is $11$ (How many distinct ways ...
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A cone has a height of $10\;\mathrm{ cm}$ and base with radius of $4\;\mathrm{ cm}$. Find the volume of the cone.

A cone has a height of $10\;\mathrm{ cm}$ and base with radius of $4\;\mathrm{ cm}$ (a)Find the volume of the cone. (b) A cone frustum is formed when you eliminate the superior part of the cone with ...
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Optimal path on the surface of cuboid doesn't visit the same face twice

There is a room in the shape of the cuboid. A spider sits somewhere on one of the faces. And there is a dead fly on one of the faces as well. The spider wants to get to the fly along the shortest path ...
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Finding an oblique pyramid with a rectangular base which admits an inscribed sphere

I want to find the apex (or locus of the apex) of a pyramid with a given (known) rectangular base, that will have an inscribed sphere. To that end, I've written a computer code, to implement the ...
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Rotating about non-centered axis in 4-d

In 3 dimensions, there is a concept of rotating about an arbitrary line that is not centered at the origin. To pull this off, we first move the origin so that it is on this line. Anywhere on the line ...
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Counterpart of axis-angle rotation matrix in 4 dimensions?

In 3-dimensional space, we have an explicit formula for the rotation matrix which will rotate about a vector $\vec{a} = [a_x, a_y, a_z]$. This is given by: $$ \begin{bmatrix} \cos\theta+a_x^2(1-\cos\...
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Prove that it isn't possible to flatten a Teserract into 2-d space.

Its certainly possible to flatten a Teserract into 3-d space (see: What does a flattened Teserract look like?). But what about 2-d space? It doesn't seem possible without having some faces fall on top ...
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Spider and fly problem on a Teserract.

In the spider and fly puzzle, there is a spider on the inside of a cuboidal room wondering how it would get to a fly on another point on the inside of the cuboid. Here, we just "flatten out" ...
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Determining the "Twist" of a Face on a Platonic Solid

I am trying to solve a 3-D design problem. I want to inscribe text on the face of a Platonic solid (whose centroid is (0,0,0)) given: a) the Cartesian coordinates of the vertices of the face; and b) ...
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Which quadrilaterals are possible as the intersection of a plane and a rectangular pyramid?

Suppose I have a pyramid with a rectangular base, and consider a plane that does not intersect the base or the apex (it only intersects the four faces incident to the apex). When the plane is parallel ...
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How many distinct ways to flatten a cube?

Think of cutting open a cubical box with the smallest possible cuts to lay it flat. A cube has 12 edges and it seems in all the possible meshes, you have to cut along 7 edges. So, the most possible ...
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What does a flattened Teserract look like?

This question is best asked with a picture: In words, we can flatten a cube into 2-d space and get a set of flattened squares like in the top right of the picture where five of the edges have stayed ...
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Fallacy in Expressions for Mass of an Elemental Part of Spheres

Fallacy in Expressions for Mass of an Elemental Part of Spheres Case-${1}$ : For a hollow sphere Let $M$ and $R$ be the mass and radius of the sphere, $O$ be it's centre and $OX$ an axis of the hollow ...
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How does this code Find the intersection point between two lines?

I've been racking my brain for a while trying to step through this. This is Unity C# code used to find the position of intersection between two lines. The full function is here Because of my usecase I ...
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The barycenter of a regular tetrahedron coincides with the center of its circumsphere

This is a self-answered question, after some playing around. I would be happy to see alternative solutions. Let $x_1,x_2,x_3,x_4 \in \mathbb{R}^3$ be the vertices of a regular tetrahedron, i.e. $|x_i-...
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The segment from a vertex of a regular tetrahedron to the center of its circumsphere is orthogonal to the opposing face

This is a self-answered question, after some playing around. I would be happy to see alternative solutions. Let $x_1,x_2,x_3,x_4 \in \mathbb{R}^3$ be the vertices of a regular tetrahedron, which lie ...
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A segment having equal angles with two equal segments is perpendicular to the connecting line

Let $a,b,c \in \mathbb{R}^3$ be unit vectors, and suppose that the angles between $a,b$, and between $a,c$ are equal. Is there an elementary, geometric, computation-free proof that $a$ is ...
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3 votes
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Conditions on tetrahedron side lengths that guarantee existence of a sphere tangent to all its edges

I recently worked on the problem of finding a sphere tangent to the edges of an irregular tetrahedron. I found that if one of the triangular faces of the tetrahedron is an isosceles triangle, then it ...
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2 votes
1 answer
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A problem involving a tetrahedron

Let $ABCD$ a tetrahedron. We know the angle $$\angle{ACB}=45^\circ$$the sum $$\overline{AD}+\overline{BC}+\frac{\overline{AC}}{\sqrt2}=90$$and that the volume is $4500.$ We also know (but I don't know ...
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Inscribed circle and square, find the equation for the perimeter of the nth shape in terms of n.

Image is of a square with side 1 which is inscribed in a circle which is inscribed in a square and so on and so forth for ever. How do I find the equation of the nth outer shape in terms of n?
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3 answers
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Height of an irregular tetrahedron with an equilateral base and lateral faces making angles $60^\circ$, $60^\circ$, $80^\circ$ with that base

An irregular tetrahedron has a base that is an equilateral triangle of side length $10$. The lateral faces make angles of $60^\circ, 60^\circ$ and $80^\circ$ with the base. Find the height of the ...
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generators of Hyperboloid

Find the point of intersection P, of the generators of opposite system drawn through the points A(acos\alpha, bsin\alpha, 0) and B(acos\beta, bsin\beta, 0) of the principle elliptical section of the ...
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How many tiles I need for pool paving [closed]

I have length, width, depth of pool and tile width. How could I calculate is it possible to do pool paving without breaking the tiles? Example: length = 5m, width = 15m, depth = 3m, tile width = 0.1m (...
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Volume of objects like hypercube / hypersphere : $V_{n}^{(m)}(r) = \dots$

I am looking for some general form of equation for calculating volume for specific geometry objects. The main idea is to find : $$ V_{n}^{(m)}(r) = \dots $$ Where: $V$ - volume of object $n$ - ...
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Intuitive way to think of the angle between planes

I'm about to teach a lesson on "The angle between planes". I'm having some question on the angle between two planes definition. Many places defines the angle between two planes as the angle ...
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How did $1/3$ comes from in the volume of tetrahedron? [duplicate]

The volume of tetrahedron is given by $$\frac{1}{3}(\text{Area of base})(\text{vertical height})$$ Similar formula is applicable for the volume of a cone. I know that a right circular cone can be ...
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How to express a cylinder as a toroid surface?

I've read that cylinders can be expressed as toroids, but the equation for aspheric surfaces can contain 2nd, 4th and higher order terms. How could a cylinder of a radius 10cm and height 30cm be ...
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Confusion regarding volume of fluid displaced by a partially immersed body

Say I have a cylindrical apparatus partially filled with a water column, the height of the column being $h$. Now I have a solid cylinder of radius smaller than the apparatus and height $H>h$ but ...
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nine points on a sphere

Show with proof that there exist $9$ points on the unit sphere (centred at the origin) so that each of the $9$ points has exactly $4$ equidistinct nearest neighbours. I found a solution to this ...
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1 vote
2 answers
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How we may express four squares whose difference is each a square in terms of (preferably solid) geometry?

The problem of finding four squares whose difference is each a square is much more exhaustive as I thought. A quest up to $2^{34}$ yields nothing. The largest almost solution found in the range up to $...
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Prove that if a convex solid is contained in another, then the surface area of the first is less or equal to the surface area of the second.

Prove that if a convex solid object $A$ is contained in another convex solid object $B$, then the surface area of $A$ is less or equal to the surface area of $B$. I already have a sketch of a proof ...
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Vector calculation seems unable to compute angle between two planes

Problem Given the cube $ABCD.EFGH$. The point $M$ is on the edge $AD$ such that $|AM|=2|MD|$. Calculate the tangent of the angle between the planes $BCF$ and $BGM$. (A) $3 \sqrt 2$ (B) $2 \sqrt 2$ (C)...
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3 votes
2 answers
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What is the volume of the largest truncated octahedron that can be inscribed in the unit sphere?

If you were to maximize the volume of a truncated octahedron while keeping it in completely inside a given sphere, what percentage of the sphere's volume would it take up? This question is an ...
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1 vote
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What is the maximal volume of the torus?

We have a $10 \, {\rm m}^2$ metal plate and we want to construct a toroidal gas tank, as shown below. Assuming that all metal is used, what must the values of inner radius $a$ and outer radius $b$ be ...
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1 vote
1 answer
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How many convex solids have the same faces?

The five platonic solids have all faces regular Polyhedra. They are also all convex. But there are other properties they satisfy. Like each vertex should have the same number of edges meeting there. ...
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2 votes
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For a fixed surface area, what is optimal shape of a boat so that it can carry the most weight?

This problem is motivated by the Penny Boat Challenge: you are given an aluminum foil and you have to create a boat out of it that can hold the most amount of pennies. I know that Archimedes' ...
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Find the volume of the solid obtained by rotating $R$ about $y=\frac{x}{5}$. [closed]

Consider the region $R$ given by $$R = \{(x,y)\in\mathbb{R}^2:(x-3)^2+(y-4)^2\leq 4 \text{ and } y\leq x\}.$$ Find the volume of the solid obtained by rotating $R$ about $y=\frac{x}{5}$. I have done ...
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1 vote
1 answer
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Volume of Trapezoidal Prism

For a plot of land of 100 m × 80 m, the level is to be raised by spreading the earth from a stack of a rectangular base 10 m × 8 m with vertical section being a trapezium of height 2 m. The top of the ...
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prove the formula for the circumradius of an isosceles tetrahedron

Prove that the circumradius of an isosceles tetrahedron (one where opposite sides are equal) with sides $a,b,c$ is given by $R= \sqrt{\frac{1}8(a^2+b^2+c^2)}$. I know how to show using vectors that ...
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1 vote
1 answer
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Find the point in a tetrahedron that minimizes the sum of distances between vertices of a tetrahedron

Tetrahedron $ABCD$ has $AD=BC, AC=BD,$ and $AB=CD$. Find the point $X$ that minimizes the sum of the distances to the four vertices (i.e. it minimizes $f(X) = AX+BX+CX+DX$). I found the following ...
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2 votes
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Find the number of different lines containing $8$ points

A $10\times 10\times 10$ grid of points consists of all points in space of the form $(i,j,k)$ where $i,j,k$ are integers between $1$ and $10$ inclusive. Find the number of distinct lines containing ...
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Derivation of prismoidal formula

The prismoidal formula is a formula used to compute for the volume of many common solids. One representation of the formula is $$ V = \frac{h}{6} (A_T+4A_M+A_B) $$ Where the height, and the areas for ...
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proof that a line XO, where X is equidistant from A,B,C is perpendicular to the plane P

Let $X$ be equidistant from points $A,B,C$, where $A,B,C$ lie in the same plane $P$ and form a (non-degenerate) triangle and $X$ is not in the plane $P$. Does it follow that $XO$ is perpendicular to $...
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2 votes
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Calculus proof for volume of a general pyramid

I want to use calculus to generalize the volume of a square pyramid to a pyramid with any base. Is the proof below correct? Are there conditions/restrictions Introduce a $z$ axis perpendicular to the ...
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Radius of sphere in which a tetrahedron can be inscribed

Find, with proof, all positive integers $N$ for which the sphere centered at the origin of radius $N$ has an inscribed regular tetrahedron whose vertices have integral coordinates. Clearly if $N=3m^2$...
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Limits on a triple integral

Find the triple integral $(x-y)dV$ of the following solid, whose limits are y=[0,2] and z=[$-x^2+1,x^2-1$] I know how to solve the integrals, but i am struggling to analyse the situation. Should I ...
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Prove that there is no convex polyhedron

Prove that there is no convex polyhedron with exactly $7$ edges Solution: We show first that for any polyhedron we have $2E \geq 3F$ and $2E \geq 3V$. The faces of the polyhedron are polygons, each ...
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Calculate coordinate of center of gravity

Question: Let it be $\Sigma$ homogeneous surface, which ia given as a graph of the function $f(x,y)=\frac{a}{\sqrt2}\cosh(\frac{x+y}{a})$ above square $[-a,a]\times[-a,a], a>0.$ Calculate ...
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Calculate moment of inertia of body $T$ around $z$-axis

I have the homogeneous body $T$, which is bounded by surfaces $x+z=1$, $y-x=1$, $y=0$ and $z=1$. Calculate moment of inertia of body $T$ around $z$-axis. I know that density of homogeneous body is $\...
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2 votes
2 answers
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Calculating the volume of the solid defined by $x^2+y^2+z^2 \leq 2a^2$ and $z \leq \frac{x^2+y^2}{a}$, with $a>0$

I have to calculate the volume of solid geometry for $a>0$ $$T=\{(x,y,z)\in \mathbb{R}; x^2+y^2+z^2 \leq 2a^2; z \leq \frac{x^2+y^2}{a}\}$$ I know that first formula is inside of sphere with ...
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