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2
votes
2answers
173 views

Why isn't the volume formula for a cone $\pi r^2h$?

So I understand that the volume formula of a cone is: $\frac{1}{3}\pi r^2h$, and when I read about how to derive this formula, it makes sense to me. Funny thing is, I'm stuck on why it ISN'T $\pi ...
2
votes
0answers
67 views

Inscribed spheres in irregular tetrahedra

Let $ABCD$ be an irregular tetrahedron, and let $K$ be the center of its inscribed sphere. Let $M$ be the center of the inscribed sphere of $KBCD$. Are $A$, $M$, $K$ necessarily collinear? I have ...
0
votes
0answers
51 views

Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we ...
0
votes
1answer
59 views

Pyramid problem - find the height of the pyramid by given edges and a side from the base

The base of a pyramid is a right triangle with the longest side = 12cm. All non base edges are also equal to 12cm. And how can I find the height of the pyramid?
3
votes
1answer
62 views

A cross-section of a pyramid

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a repost from this. Can someone help me? Thanks in advance.
2
votes
1answer
38 views

Computing Solid Geometry from Surfaces - CAD

I'm an engineer and I recently took a course at my local university about CAD curves and surfaces. I understand how NURBS surfaces work and how to generate the complex surfaces that models consist of. ...
0
votes
1answer
55 views

Intersection of a plane and a surface of revolution

I'm am stuck on the following problem: I have the equation of a curve in the plane $(x,z)$: $z=f(x)$. I build a surface of revolution in the space rotating this curve around the $z$ axis. I need to ...
2
votes
1answer
78 views

Regular pyramid, centre of circumscribed ball=centre of inscribed ball

Prove that if the sum of plane angles in the vertex of a regular pyramid equals $180 ^{\circ}$, then in this pyramid the centers of the inscribed and circumscribed balls are equal. Could you help me ...
2
votes
1answer
43 views

Find radius of sphere

Imagine eight spheres of radius 1 that are at $(\pm1,\pm1,\pm1)$. Place sphere A with its center at the origin externally tangent to all of the other spheres. Then place sphere B externally tangent to ...
4
votes
2answers
110 views

Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?

Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra: http://en.wikipedia.org/wiki/Platonic_solid http://en.wikipedia.org/wiki/Archimedean_solid ...
0
votes
0answers
31 views

volume swept by a triangle

I have two triangles defined in space, say $(a,b,c)$ and $(a',b',c')$, where the second one is the outcome of a linear transform of the first. The lines connecting corresponding vertices ($(a-a')$, ...
1
vote
1answer
72 views

Solid Trigonometry

V ABCD is a pyramid with the vertex V situated perpendicularly above the centre of the square base ABCD. If $\theta$ is the angle between the edge VA and the base, and $\phi$ is the angle between the ...
6
votes
1answer
111 views

Platonic solids and charged particles

It is known that there are five Platonic solids: If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will ...
1
vote
1answer
84 views

How can the tension force be computed to test if a shape is moving or not?

Source Given the coordinates of n 3D joints (1kg each) connected by m rods. Assume rods have zero mass and joints with z=0 are fixed to the ground while others are free to move, will the shape be ...
3
votes
1answer
106 views

How to compute force on joints of a 3D structure of balls connected by rods?

Source Given the coordinates of n 3D joints (1kg each) connected by m rods. Assume rods have zero mass and joints with z=0 are fixed to the ground while others are free to move, will the shape be ...
0
votes
1answer
88 views

Triangle inside a cylinder, surface times circumference - hard to visulize question

The surface of the triangle is $X$ and the circumfrence of the base of the cylinder is $Y$. What is the volume of the cylinder ? The answer is $XY$ but why ? If you'll take the ...
2
votes
1answer
307 views

Solids of Revolution - Negative volume?

So I'm to find the volume of the solid formed by the area trapped between $y = x$ and $y = \sqrt{x}$, rotated about $y = 1$ The two curves cross at y = 0 and y = 1 so they will be the upper/lower ...
-1
votes
1answer
40 views

locus sections and circles--symmetry

A. Let L = {(x,y,z)|the distance from (x,y,z) to the y-axis is 6}. Describe what shape L is. So far, I have that it's a circle, but I'm not sure how to describe it fully. Would it be a circle that ...
1
vote
2answers
264 views

Use cylinder's formula for frustum (conical frustum)

I know that frustum(conical) has a formula for its volume,i.e. $\frac1 3\pi h(r^2+R^2+rR)$, but why can't we place the average of two radii into cylinder's formula: $\pi(\frac{r+R}2)^2(h)$? I need ...
0
votes
1answer
250 views

How will I get the volume and surface area of the sphere?

If a rectangular solid have edges, 4 cm, 5 cm and 7 cm, is inscribed in a sphere. How will I get the volume and surface area of the sphere? Do I need to get the volume of the rectangular solid and ...
6
votes
1answer
162 views

Sum of radii of exspheres

I am interested in finding some results for tetrahedron that would be analoguous to known results for triangle. In triangle with circumradius R and inradius r, if we consider the excircles $r_i$, then ...
0
votes
1answer
193 views

Ellipse Tangents in 3D

I know that we can find the tangent of the ellipse in 2D by taking the derivative of the equation defining the ellipse. But I'm little bit confused about finding the ellipse tangent in 3D. Where the ...
0
votes
1answer
65 views

Intersected cone, a practical problem

This is the maths representation of a problem which I have from the practice. We have an interested cone with a diameter of the base $d$, height $h$ and the angle $\alpha$ as shown on the drawing. A ...
0
votes
2answers
63 views

Point of tangency between plane and sphere

How can one find the point of tangency between a plane and a sphere in $\mathbb{R}^3$? The equations of the plane and the sphere are $x + y + z - 5 = 0$ and $(x-1)^2 + (y-2)^2 + (z+1)^2 = 3$ ...
1
vote
1answer
87 views

Find the area of the Grayed triangle Given the following Figure

Can you help me find the area of the gray triangle in the given figure. I'm having a hard time finding the base value of the triangle, I've managed to find the sides for the big triangle but not ...
0
votes
1answer
582 views

Line Drawing Using Bresenham Algorithm

Indicate which raster locations would be chosen by Bersenham’s algorithm when scan converting a line from screen co-ordinates (1,1) to (8,5). First the straight values (initial values) must be found ...
0
votes
1answer
119 views

vectors in 3D space and Right-Hand Rule

Suppose we have three vectors in 3D space. My questions are: How we check if these vectors are satisfy the right-hand rule or not. I know that it's possible to make the three vectors satisfy the ...
2
votes
3answers
130 views

Generalization of angle bisector to tetrahedron

Let $I$ be the center of the inscribed sphere of a tetrahedron $ABCD$ and let $I_A$ be the length of the line passing through $I$ from the vertex $A$ to the opposite face. I am looking for a formula ...
2
votes
0answers
22 views

The exact type of my 3d model

I have reconstructed vertical features (hole like objects lie on a vertical face) lie on two connected faces. To understand the situation, I say I have 2 walls with many windows and doors on ...
0
votes
1answer
111 views

Perspective projection onto y/z plane?

On wikipedia there is an article on 3d perspective projection onto the x/y plane. http://en.wikipedia.org/wiki/3D_projection#Perspective_projection How do I project onto the y/z plane? If i have a ...
1
vote
1answer
148 views

Icosahedron and dual dodecahedron coordinates and rotations

I'm trying to build a 20 sided die in Actionscript 3, like one in the picture. I figure the best way would be to make it out of 20 equilateral triangles rotated in 3D space. The vertices of the ...
1
vote
1answer
24 views

calculate size of faces to have similar volume of platonic solids

I need to build some prototype physical shapes of the 5 platonic solids. But the problem is that I want to make them all about the same size, so how to calculate how big should the faces of my ...
2
votes
3answers
55 views

what will be the angle at the centre?

Taken a tetrahedron of same edges, a point is taken inside it which is equidistant from all $4$ vertices, i.e if a sphere is made taking it as a centre, all the vertices will be on the sphere, now ...
2
votes
1answer
157 views

Disk and washer problem

I need to find the volume of the solid obtained by rotating the region bounded by the following functions: $xy = 1, y = 0, x = 1, x = 2;$ about $x = -1$ I think the outer radius of the washer is ...
5
votes
3answers
146 views

Which solids are characterized by their orthographic projections?

If I know the orthographic projections of a given solid in Euclidean 3-space onto the $xy$, $xz$ and $yz$ planes, under which circumstances can I reconstruct the solid based on that information alone? ...
1
vote
0answers
53 views

Finding an angle in a straight pyramid

We have a straight pyramid with a square ABCD as its base and apex S. We're given the pyramid's height 8 and the angle 48 deg. between SA and SC. I've already managed to calculate the pyramid's volume ...
2
votes
0answers
131 views

Finding an angle in a pyramid

We have a straight pyramid with a square ABCD as its base and apex S. We're given the pyramid's height 8 and the angle 48 deg. between SA and SC. I've already managed to calculate the pyramid's volume ...
13
votes
1answer
940 views

Is there a dissection proof of the Pythagorean Theorem for tetrahedra?

Of the many nice proofs of the Pythagorean theorem, one large class is the "dissection" proofs, where the sum of the areas of the squares on the two legs is shown to be the same as the area of the ...
5
votes
1answer
240 views

Volume of a sphere with three holes drilled in it.

Suppose that the sphere $x^2+y^2+z^2=9$ has three holes of radius $1$ drilled through it. One down the $z$-axis, one along the $x$-axis, and one along the $y$-axis. What is the volume of the resulting ...
3
votes
0answers
113 views

Angle between two planes, finding the second plane

I've got a plane given by $x-y+z=0$ and a line given by $r=(0,0,a)+t(1,1,b)$. How do I find another plane that contains the line and intersects the other plane at an angle of $45^\circ$? Is ...
0
votes
1answer
103 views

Eigenvectors for the equation of the second degree and right-hand rule

I'm trying to find the Eigenvectors for the equation of the second degree (for example Elliptic cone). The estimated values $V_1$, $V_2$ and $V_3$ must satisfy the right-hand rule. How can we verify ...
4
votes
1answer
242 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
1
vote
1answer
357 views

Find the volume of the solid about y-axis, i.e. 1/(x^2+3x+2), bounds x=0, x=1

find intersection points: 1/(x^2+3x+2)=1, so x=(sqrt(5)-3)/2, x= -(sqrt(5)+3)/2 thus we have: V(volume)= Pi*integral((1)^2-(1/(x^2+3x+2)^2) from x= -(sqrt(5)+3)/2, x= (sqrt(5)-3)/2 ...
1
vote
2answers
115 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
0
votes
1answer
109 views

Correcting plane parameters with the fixed azimuth angles

I am trying to reconstruct specific 3d objects such as cubes, pyramids and so on. For this, i am using point cloud data and then fitting planar surfaces for the segmented point patches. Planes ...
2
votes
0answers
45 views

How to fit a cuboid into a polyhedra?

I have multiple points which create a solid (polyhedra). And now I want to place a cuboid inside this solid in a way that it uses the maximum amout of space inside. Are there any solutions for this ...
2
votes
1answer
297 views

solid angle of polyhedron

I have an interesting thought when I draw polygon and 3D polyhedron. My question is: Can I know the number of Face, Edge, and Vertex from a given 'space angle constraint'? For example, a vertex a cube ...
1
vote
3answers
326 views

Surface area comparison of a solid cut in half

This was just a thought I had while driving this morning, no particular application. If you make a straight line cut through a solid object such that the resulting two pieces have the same volume, ...
2
votes
1answer
229 views

Prove that $S$ is a sphere.

Let $S\subset {\mathbb{R}}^3$ with the following properties: $1.$ For any line $l$: $|l\cap S|=2$ or $|l\cap S|=1$ or $|l\cap S|=0$ $2.$ For any plane $P$ $P\cap S=\text{circle}$ or $|P\cap S|=1$ ...
0
votes
1answer
393 views

Numerical computation of surface curvature

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is \begin{equation} C = \frac{y''}{(1+y'^{2})^{3/2}} \end{equation} and it is easy to estimate the curvature numerically for given ...