# What is Inscribed: Definition and 89 Discussions

In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon (or a sphere or ellipsoid inscribed in a convex polyhedron) is tangent to every side or face of the outer figure (but see Inscribed sphere for semantic variants). A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.
Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon. A polygon inscribed in a circle is said to be a cyclic polygon, and the circle is said to be its circumscribed circle or circumcircle.
The inradius or filling radius of a given outer figure is the radius of the inscribed circle or sphere, if it exists.
The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces.
For an alternative usage of the term "inscribed", see the inscribed square problem, in which a square is considered to be inscribed in another figure (even a non-convex one) if all four of its vertices are on that figure.

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1. ### B A Simpler Way to Find the Shaded Area?

Consider the following scenario: Given that points ##M## and ##N## are the midpoints of their respective line segments, what would be the fastest way to determine what percentage of the squares total area is shaded purple? I managed to determine that the purple shaded area is ##5\text{%}##...
2. ### Find the area of the shaded region in the inscribed circle on square

Find the solution here; Find my approach below; In my working i have; ##A_{minor sector}##=##\frac {128.1^0}{360^0}×π×5×5=27.947cm^2## ##A_{triangle}##=##\frac {1}{2}####×5×5×sin 128.1^0=9.8366cm^2## ##A_3##=##\frac {90^0}{360^0}####×π×10×10##=##78.53cm^2## ##A_{major...
3. ### MHB ASVAB circle and inscribed rectangle area problem

Rectangle ABCD is inscribed in the circle shown. If the length of side $\overline{AB}$ is 5 and the length of side $\overline{BC}$ is 12 what is the area of the shaded region? $a.\ 40.8\quad b.\ 53.1\quad c\ 72.7\quad d \ 78.5\quad e\ 81.7$ well to start with the common triangle of 12 5...
4. ### Optimization problem - right circular cylinder inscribed in cone

Please I do not want the answer, I just want understanding as to why my logic is faulty. Included as an attachment is how I picture the problem. My logic: Take the volume of the cone, subtract it by the volume of the cylinder. Take the derivative. from here I can find the point that the cone...
5. ### MHB Minimum Length of Longest Side in Inscribed Triangle

In triangle ABC, ∠C = 90 degrees, ∠A = 30 degrees and BC = 1. Find the minimum length of the longest side of a triangle inscribed in triangle ABC (that is, one such that each side of ABC contains a different vertex of the triangle).
6. ### Area remaining of a quarter-circle deprived of these 3 inscribed circles

Summary:: Calculate the percentage of area remaining when a quarter-cirlce is deprived of 1 large circles and 2 smaller circles. Hi, I'm not sure if this is the right subforum for this question but it seemed to be the one that fit the best. Please consider the following diagram: Before...
7. ### MHB Largest Quadrilateral inscribed in Scalene Triangle

I have a scalene triangle: A: 75.04 B: 66.9 C: 41.13 The first thing I need to do is move just lines A and C in towards each other .5 and recalculate all sides. Then I need to inscribe the largest quadrilateral that will fit while having one side being no shorter than 7.5, with the entire...
8. ### Find this angle given the triangle's Orthocenter

##AD## is diameter, thus ##\angle ACD = \angle ABD = 90^\circ##. Also ##HBDC## is a parallelogram because ##HC||BD, HB||CD##. It seems useless and I don't know how to continue. Thank you in advance!
9. ### Angles inscribed in circles part 1

Homework Statement OK, I am new to these kinds of problems and I am trying to learn the appropriate properties but they are proving somewhat difficult for me... I hope I am going in the right direction. Homework Equations [/B] The first problem corresponds to the figure with 'Rep' in the...
10. ### Triangle inscribed in a circle

Homework Statement [/B] In a circle with center S, DB is the diameter. The line AC goes 90 degrees from the center point M of the line SB. " What type of triangle is ACD? 2. Homework Equations The Attempt at a Solution I can see it is an equilateral triangle, but do not know how to explain...
11. ### MHB AO+BO+CO≥6r where r is the radius of the inscribed circle

From the entrance examinations to Ghana University ,from high school, i got the following problem: If O is the center of the inscribed circle in an ABC trigon,then prove that: AO+BO+CO\geq 6r where r is the radius of the inscribed circle.
12. ### MHB What is the Sum of Lengths for a Regular n-gon Inscribed in a Unit Circle?

Let $S_n$ be the sum of lengths of all the sides and all the diagonals of a regular $n$-gon inscribed in a unit circle. (a). Find $S_n$. (b). Find $$\lim_{{n}\to{\infty}}\frac{S_n}{n^2}$$
13. ### Maximum area for inscribed cylinder

Homework Statement Inscribe in a given cone, the height h of which is equal to the radius r of the base, a cylinder (c) whose total area is a maximum. Radius of cylinder is rc and height of cylinder is hc. Homework Equations A = 2πrchc + 2πrc2 The Attempt at a Solution r = h ∴ hc = r - rc A =...
14. ### MHB What is the total area of the infinite number of inscribed squares?

Given a circle (radius $R$) with an inscribed square. Now inscribe a new circle in the square and then again a new square in the new circle etc. What is the total area of the infinite number of inscribed squares?
15. ### Circle inscribed in a triangle exercise

Homework Statement In the drawing you can see a circumference inscribed in the triangle ABC (See the picture in the following link). Calculate the value of X https://goo.gl/photos/CAacV2dJbUrywfXv92. The attempt at a solution It seems I found a solution for this exercise with the help of a...
16. ### MHB Inscribed Angles: A Guide to Identifying and Understanding

I've been having trouble identifying these inscribed angles for a while. I know the theorem that goes with this topic but I'm unsure how it applies :c.
17. ### Portion of Altitude of a Triangle Inscribed in a Circle

Homework Statement In the figure Q image2.jpeg (attached), equilateral triangle ABC is inscribed in circle O, whose radius is 4. Altitude BD is extended until it intersects the circle at E. What is the length of DE? Solution figure is attached. They formed a right angled triangle & calling it...
18. ### MHB Maximizing volume of cone inscribed within cone

A right circular cone is inscribed inside a larger right circular cone with a volume of 150 cm3. The axes of the cones coincide and the vertex of the inner cones touches the center of the base of the outer cone. Find the ratio of the heights of the cones that maximizes the volume of the inner...
19. ### MHB Inscribed circle in the triangle

In the triangle a point I is a centre of inscribed circle. A line AI meets a segment BC in a point D. A bisector of AD meets lines BI and CI respectively in a points P and Q. Prove that heights of triangle PQD meet in the point I. I've tried to show that sides of triangle PQD are parallel to...
20. ### The radius of a circle inscribed in 2 triangles

Hi guys, was wondering if anyone could help me solve this problem. Thanks!
21. ### Side lengths of inscribed triangle

Homework Statement The corners A, B and C of a triangle lies on a circle with radius 3. We say the triangle is inscribed in the circle. ∠A is 40° and ∠B is 80°. Find the length of the sides AB, BC and AC. Homework EquationsThe Attempt at a Solution I found out the arc AB is 2π, arc BC is 4π/3...
22. ### Finding Circle Circumference from Inscribed N-Sided Polygen

The perimeter P of a regular polygon of n sides inscribed in a circle of radius r is given by P = 2nr sin (180^o / n). I was curious whether it's possible to approximate the circumference of a circle by taking the limit as n goes to infinity of the above perimeter equation is some way? Thank-you
23. ### Maximum area of a triangle inscribed in another triangle?

Homework Statement [/B] Hello! I have this question which I don't quite know how to solve... ABC is an equilateral triangle - the length of its sides equal to (a). DE is parallel to BC 1. What length should DE be to achieve the largest possible area of triangle BDE? 2. What length should DE...
24. ### Equilateral triangle ABC is inscribed in a circle

Homework Statement Can someone help me solve this, and teach me how to solve such problems in future? An equilateral triangle ABC is inscribed in a circle . Point D lies on a shorter arc of a circle BC. Point E is symmetrical the point B relating to the line CD . Prove that the points A, D , E...
25. ### MHB Minimizing Isosceles triangle with a circle inscribed

Find the smallest possible area of an isosceles triangle that has a circle of radius $r$ inside it. I cannot seem to find the relationship between the circle and triangle. Any hints? I'm thinking similar triangles, but I want to know if they're any other approaches before I try that.
26. ### MHB Radius of Sphere Inscribed in Square Pyramid: Michelle's Q&A

Here is the question: I have posted a link there to this thread so the OP can view my work.
27. ### MHB Maximizing Area of Inscribed Rectangle - Yahoo Answers

Here is the question: I have posted a link there to this thread so the OP can view my work.
28. ### Circle Inscribed in a Parabola?

Homework Statement Find the largest circle centered on the positive y-axis which touches the origin and which is above y=x^2 Homework Equations equation of a circle: r^2=(x-a)^2+(y-b)^2 equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2 equation of a parabola: y=x^2 The...
29. ### Optimization - rectangle inscribed in a right triangle

Homework Statement A rectangle is to be inscribed in a right triangle having sides 3 cm, 4 cm and 5 cm, as shown on the diagram. Find the dimensions of the rectangle with greatest possible area. Homework Equations 1. x^{2}+y^{2}=w^{2} in terms of w=\sqrt{x^{2}+y^{2}} 2...
30. ### MHB Prove Triangle Inscribed in Semicircle Is Right Angle | Arundev Answers

Here is the question: I have posted a link there to this thread so the OP can view my work.
31. ### MHB A quadrilateral inscribed in a semicircle

Let $PQRS$ be a quadrilateral inscribed in a semicircle with diameter $PS=x$. If $PQ=a$, $QR=b$, $RS=c$, then prove that $x^3-(a^2+b^2+c^2)x-2abc=0$.
32. ### MHB Khegan McLane's Math Problem: Rectangle Inscribed Between Parabola & X-Axis

Here is the question: I have posted a link there to this thread so the OP can see my work.
33. ### MIN/MAX area of a rectangle inscribed in a rectangle

http://bp3.blogger.com/_4Z2DKqKRYUc/Rnz_BgODzFI/AAAAAAAAAIw/uj_cVfPI8D4/s1600-h/Img_6-23-07_Blog.jpg Could someone help me to understand how can I figure it out, how can I create a formula for finding min/max area of a rectangle inscribed in a rectangle, defined by given width and height. Also...
34. ### MHB Max Volume of Cylinder Inscribed in a Cone: Dimensions?

Here is the question: I have posted a link there to this topic so the OP can see my work.
35. ### Optimization of Area; Inscribed Rectangles

Homework Statement http://i.minus.com/jbkHm5oH1LfQ1k.png Homework Equations The area of a rectangle is its base times its height. The Attempt at a Solution The rectangle is inscribed. Its area is 2xy. I can substitute in the equation of the semicircle to get rid of the y-term...
36. ### MHB Find Max Volume of Cylinder Inscribed in Cone

Thanks again to those who participated in last week's POTW! Here's this week's problem! ----- Problem: A right circular cylinder is inscribed in a cone with height $h$ and base radius $r$. Find the largest possible volume of such a cylinder. -----
37. ### Finding diagonal of inscribed rectangle

Homework Statement A quadrant contains an inscribed rectangle ABCD. Given the distance marked: CD=3m , what is length of AD? Homework Equations Area of circle = pi*r^2 Pythagorean 's theorem : a^2=b^2+c^2 The Attempt at a Solution We can draw diagonal from C to B similar to...
38. ### MHB Inscribed and circumscribed quadrilateral

I would like to discuss the following problem. The quadrilateral ABCD is inscribed into a circle of given radius R. And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral KLMN such that S_{ABCD}=3S_{KLMN}. Also \gamma is the angle between...
39. ### Geometry: Triangle with a Circumscribed and Inscribed Circle

Homework Statement What is the area of a right triangle whose inscribed circle has radius 3 and whose circumscribed circle has a radius 8? Homework Equations The diameter must be the hypotenuse of the circle The Attempt at a Solution The answer is 57, but I do not know the...
40. ### A rather difficult problem: deltoid inscribed into an ellipse

I'm a collage teacher and I've found a very hard problem in one of my math classrooms' textbooks. It was firstly proposed as problem n. 9, back in 1995, in the "Annual Iowa Collegiate Mathematics Competition". Link is here (no solution file available in the site for that year). The text is...
41. ### MHB Equations of Sides of Square Inscribed in Circle

Find the equations of the sides of square inscribed in the circle $3(x^2+y^2)=4$, one of whose sides is parallel to the line $x-y=7$.
42. ### Triangle inscribed on circle proof I am missing something

Triangle inscribed on circle proof...I am missing something :( Homework Statement I have provided a link to the problem below http://imageshack.us/a/img854/4143/photo1lsd.jpg I need to prove AE is an altitude on this proof Homework Equations all radii are congruent, cpctc, ASA...

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44. ### Proof: circumference of circle inscribed in a triangle < perimeter of the triangle

Homework Statement A circle is inscribed in a triangleHere is a picture Picture of circle inscribed in triangle, not necessarily to scaleWhich is larger: the circumference of the circle, or the perimeter of the triangle? Homework EquationsC=∏D (D=diameter of the circle, C=circumference of...
45. ### Square Inscribed in a Square: Maximizing Distance Between Vertices

Homework Statement A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square. Homework Equations The Attempt at a Solution I have a question. Can a square be inscribed in...
46. ### Radius of Inscribed Circle in a Quadrant of a Circle

Homework Statement Find the radius of a circle inscribed in a quadrant of a circle with radius 5 Homework Equations The Attempt at a Solution I worked this but I'm not sure if its correct. I looked at the first quadrant so a quarter of a circle with radius 5. I drew the radius...
47. ### Inscribed sphere - Kepler Conjecture

Newbie to the forum here. Hoping y'all can help with something that's been bugging me for a while now. I would like to know the relationship between two characteristic radii in a close packing of equal spheres. The first radius of interest is that of the equal sphere's themselves (r1). The...
48. ### 2 Inscribed Angle Geometry Problems

I want to find x for (1) and x and y for (2). I am not sure how to put the images directly into the thread so I apologize if you do not like having to click on them. On the first one, I do not know how we can find this without knowing that XW is a diameter (it is not given as one in the...
49. ### Find the maximum value of a rectangular box that can be inscribed in an ellipsoid

"Find the maximum value of a rectangular box that can be inscribed in an ellipsoid.." Homework Statement Find the maximum value of a rectangular box that can be inscribed in an ellipsoid x^2 /4 + y^2 /64 + z^2 /81 = 1 with sides parallel to the coordinate to the coordinate axes...
50. ### Circle Inscribed in Triangle: Area Ratios with Inscribed Circle Tangents

Homework Statement Consider a triangle ABC, where angle A = 60o. Let O be the inscribed circle of triangle ABC, as shown in the figure. Let D, E and F be the points at which circle O is tangent to the sides AB, BC and CA. And let G be the point of intersection of the line segment AE and the...