INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT

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Bunuel

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 04:26

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INSCRIBING POLYGONS AND CIRCLES

By Karishma, Veritas Prep.

Above we looked at regular and irregular polygons. Now, let’s try to understand how questions involving one figure inscribed in another are done. The most common example of a figure inscribed in another is a polygon inscribed in a circle or a circle inscribed in a polygon. Let’s see the various ways in which this can be done.

To inscribe a polygon in a circle, the polygon is placed inside the circle so that all the vertices of the polygon lie on the circumference of the circle.

There are a few points about inscribing a polygon in a circle that you need to keep in mind:
– Every triangle has a circumcircle so all triangles can be inscribed in a circle.

– All regular polygons can also be inscribed in a circle.

– Also, all convex quadrilaterals whose opposite angles sum up to 180 degrees can be inscribed in a circle.

There are also a few points about inscribing a circle in a polygon that you need to keep in mind:
– All triangles have an inscribed circle (called incircle). When a circle is inscribed in a triangle, all sides of the triangle must be tangent to the circle.

– All regular polygons have an inscribed circle.

– Most other polygons do not have an inscribed circle.

A simple official question will help us see the relevance of these points:
Question: Which of the figures below can be inscribed in a circle?

(A) I only
(B) III only
(C) I & III only
(D) II & III only
(E) I, II & III

Solution: I think it will suffice to say that the answer is (C).
This question is discussed HERE.

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 04:36

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CIRCLES AND INSCRIBED REGULAR POLYGON RELATIONS

By Karishma, Veritas Prep.

Let’s figure out the relations between the sides of various inscribed regular polygons and the radius of the circle.

We will start with the simplest regular polygon – an equilateral triangle. We will use what we already know about triangles to arrive at the required relations.

Look at the figure given below. AB, BC and AC are sides (of length ‘a’) of the equilateral triangle. OA, OB and OC are radii (of length ‘r’) of the circle.

The interior angles of an equilateral triangle are 60 degrees each. Therefore, angle OBD is 30 degrees (since ABC is an equilateral triangle, BO will bisect angle ABD). So, triangle BOD is a 30-60-90 triangle.

As discussed in your geometry book, the ratio of sides in a 30-60-90 triangle is $1:\sqrt{3}:2$ therefore, a/2 : r = $\sqrt{3}:2$ or $a:r = \sqrt{3}:1$

Side of the triangle = $\sqrt{3}$ * Radius of the circle

You don’t have to learn up this result. You can derive it if needed. Note that you can derive it using many other methods. Another method that easily comes to mind is using the altitude AD. Altitude AD of an equilateral triangle is given by $\frac{\sqrt{3}}{2}*a$. The circum center is at a distance 2/3rd of the altitude so AO (radius) = $(\frac{2}{3})*\frac{\sqrt{3}}{2}*a = \frac{a}{\sqrt{3}}$

Or side of the triangle = $\sqrt{3}$ * radius of the circle

Let’s look at a square now.

AB is the side of the square and AO and BO are the radii of the circle. Each interior angle of a square is 90 degrees so half of that angle will be 45 degrees. Therefore, ABO is a 45-45-90 triangle. We know that the ratio of sides in a 45-45-90 triangle is $1:1:\sqrt{2}$.

$r:a = 1: \sqrt{2}$

Side of the square = $\sqrt{2}$*Radius of the circle

Again, no need to learn up the result. Also, there are many methods of arriving at the relation. Another one is using the diagonal of the square. The diagonal of a square is $\sqrt{2}$ times the side of the square. The radius of the circle is half the diagonal. So the side is the square is $\sqrt{2}$*radius of the circle.

The case of a pentagon is more complicated since it needs the working knowledge of trigonometry which is beyond GMAT scope so we will not delve into it.

We will look at a hexagon though.

Notice that the interior angle of a regular hexagon is 120 degrees so half of that will be 60 degrees. Therefore, both angles OAB and OBA will be 60 degrees each. This means that triangle OAB is an equilateral triangle with all angles 60 degrees and all sides equal. Hence,

Side of the regular hexagon = Radius of the circle.

The higher order regular polygons and more complicated and we will not take them up. We will discuss a circle inscribed in a polygon in the next post.

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 04:47

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AND NOW THE OTHER WAY

By Karishma, Veritas Prep.

In this post will work with circles inscribed in regular polygons.

We begin by considering an equilateral triangle whose each side is of length ‘a’. Recall that every triangle has an incircle i.e. a circle can be inscribed in every triangle. The diagram given below shows the circle of radius ‘r’ inscribed in an equilateral triangle.

How can we find the relation between ‘r’ and ‘a’? Every angle of an equilateral triangle is 60 degrees. Since it is an equilateral triangle, due to the symmetry, angle OBD = angle OBA = 30 degrees. So we see that triangle BOD is a 30-60-90 triangle. So the ratio of the sides OD:BD:OB = $1: \sqrt{3} = r : \frac{a}{2}$.

Therefore, $a = 2\sqrt{3} * r$

Side of the triangle = $2\sqrt{3}$ * Radius of the circle

As discussed last week, there are many other methods of getting this result. We can use the altitude method.

Altitude of an equilateral triangle is given by $\frac{\sqrt{3}}{2}*a$. The incenter is at a distance 2/3rd of the altitude so OD (radius) = $(\frac{1}{3})*\frac{\sqrt{3}}{2}*a = \frac{a}{2\sqrt{3}}$

Or Side of the triangle = $2\sqrt{3}$ * Radius of the circle

Now we will look at a square.

The figure itself shows us that r = a/2

Side of the square = 2 * Radius of the circle

There is no need to delve deeper into it. Though, here is something for you to think about: Can you have a circle inscribed in a rectangle?

Now let’s consider a circle inscribed in a regular hexagon.

We know that the interior angle of a regular hexagon is 120 degrees. OA will bisect that angle making angle OAD = 60 degrees. Since AB is tangent to the circle, OD will be perpendicular to AB. Hence OAD is a 30-60-90 triangle. Therefore, $\frac{a}{2} : r = 1: \sqrt{3}$

Hence, $a = \frac{2r}{\sqrt{3}}$

Side of the hexagon = $\frac{2}{\sqrt{3}}$ * Radius of the circle

Again, remember, you are not expected to ‘know’ these results so don’t try to learn them up. You can always derive any relation you want once you know some basic tricks. The intent of these posts is to familiarize you with those tricks.

In the next post, we will look at some interesting Geometry questions based on these concepts!

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 05:05

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QUESTIUONS ON POLYGONS INSCRIBED IN CIRCLES

By Karishma, Veritas Prep.

In this post, I have two questions for you – both on polygons inscribed in a circle. You must go through the previous post based on this topic before trying these questions.

Question 1: Four points that form a polygon lie on the circumference of the circle. What is the area of the polygon ABCD?
Statement I: The radius of the circle is 3 cm.
Statement II: ABCD is square.

Solution:
Notice that you have been given that angles B and D are right angles. Does that imply that the polygon is a square? No. You haven’t been given that the polygon is a regular polygon. The diagonal AC is the diameter since arc ADC subtends a right angle ABC. Hence arc ADC and arc ABC are semi-circles. But the sides of the polygon (AB, BC, CD, DA) may not be equal. Look at the diagram given below:

Statement I: The radius of the circle is 3 cm.
This statement alone is not sufficient. Look at the two figures given above. The area in the two cases will be different depending on the length of the sides. Just knowing the diagonal AC is not enough. Hence this statement alone is not sufficient.

Statement II: ABCD is square.
This tells us that the first figure is valid i.e. the polygon is actually a square. But this statement alone doesn’t give us the measure of any side/diagonal. Hence this statement alone is not sufficient.

Using both statements together, we know that ABCD is a square with a diagonal of length 6 cm. This means that the side of the square is $\frac{6}{\sqrt{2}}$ cm giving us an area of $(\frac{6}{\sqrt{2}})^2 = 18$ cm^2.

Answer (C). This question is discussed HERE.

Let’s look at a more complicated question now.

Question 2: A regular polygon is inscribed in a circle. How many sides does the polygon have?
Statement I: The length of the diagonal of the polygon is equal to the length of the diameter of the circle.
Statement II: The ratio of area of the polygon to the area of the circle is less than 2:3.

Solution:
In this question, we know that the polygon is a regular polygon i.e. all sides are equal in length. As the number of sides keeps increasing, the area of the circle enclosed in the regular polygon keeps increasing till the number of sides is infinite (i.e. we get a circle) and it overlaps with the original circle. The diagram given below will make this clearer.

Let’s look at each statement:

Statement I: The length of one of the diagonals of the polygon is equal to the length of the diameter of the circle.
Do we get the number of sides of the polygon using this statement? No. The diagram below tells you why.

Regular polygons with even number of sides will be symmetrical around their middle diagonal and hence the diagonal will be the diameter. Hence the polygon could have 4/6/8/10 etc sides. Hence this statement alone is not sufficient.

Statement II: The ratio of area of the polygon to the area of the circle is less than 2:3.

Let’s find the fraction of area enclosed by a square.

In the previous post we saw that
Side of the square = $\sqrt{2}$ * Radius of the circle
Area of the square = $Side^2 = 2*Radius^2$
Area of the circle = $\pi*Radius^2 = 3.14*Radius^2$
Ratio of area of the square to area of the circle is 2/3.14 i.e. slightly less than 2/3.

So a square encloses less than 2/3 of the area of the circle. This means a triangle will enclose even less area. Hence, we see that already the number of sides of the regular polygon could be 3 or 4. Hence this statement alone is not sufficient.

Using both statements together, we see that the polygon has 4/6/8 etc sides but the area enclosed should be less than 2/3 of the area of the circle. Hence the regular polygon must have 4 sides. Since the area of a square is a little less than 2/3rd the area of the circle, we can say with fair amount of certainty that the area of a regular hexagon will be more than 2/3rd the area of the circle. But just to be sure, you can do this:

Side of the regular hexagon = Radius of the circle

Area of a regular hexagon = 6*Area of each of the 6 equilateral triangles = $6*(\frac{\sqrt{3}}{4})*Radius^2 = 2.6*Radius^2$

2.6/3.14 is certainly more than 2/3 so the regular polygon cannot be a hexagon. The regular polygon must have 4 sides only.

Answer (C). This question is discussed HERE.

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 05:20

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QUESTIONS ON CIRCLES INSCRIBED IN POLYGONS

By Karishma, Veritas Prep.

Above we looked at questions on polygons inscribed in a circle. This week, let’s look at questions on circles inscribed in regular polygons. As noted earlier, it’s important to keep in mind that regular polygons are symmetrical figures. You need very little information to solve for anything in a symmetrical figure.

Question 1: A circle is inscribed in a regular hexagon. A regular hexagon is inscribed in this circle. Another circle is inscribed in the inner regular hexagon and so on. What is the area of the tenth such circle?

Statement I: The length of the side of the outermost regular hexagon is 6 cm.
Statement II: The length of a diagonal of the outermost regular hexagon is 12 cm.

Solution: Thankfully, in DS questions, we don’t need to calculate the answer. We just need to establish the sufficiency of the given data. Note that we have found that there is a defined relation between the sides of a regular hexagon and the radius of an inscribed circle and there is also a defined relation between the radius of a circle and the side of an inscribed regular hexagon.

When the circle is inscribed in a regular hexagon,

Radius of the inscribed circle = $\frac{\sqrt{3}}{2}$* Side of the hexagon

When a regular hexagon is inscribed in a circle,

Side of the inscribed regular hexagon = Radius of the circle

So all we need is the side of any one regular hexagon or the radius of any one circle and we will know the length of the sides of all hexagons and the radii of all circles.

Statement I: The length of the side of the outermost regular hexagon is 6 cm.
If length of the side of the outermost regular hexagon is 6 cm, the radius of the inscribed circle is $(\sqrt{3}/2)*6 = 3\sqrt{3}$ cm

In that case, the side of the regular hexagon inscribed in this circle is also $3\sqrt{3}$ cm. Now we can get the radius of the circle inscribed in this second hexagon and go on the same lines till we reach the tenth circle. This statement alone is sufficient.

Statement II: The length of a diagonal of the outermost regular hexagon is 12 cm.

Note that a hexagon has diagonals of two different lengths. The diagonals that connect vertices with one vertex between them are smaller than the diagonals that connect vertices with two vertices between them. Length of AC will be shorter than length of AD. Given the length of a diagonal, we do not know which diagonal it is. Is AC = 12 or is AD = 12? The length of the side will be different in the two cases. So this statement alone is not sufficient.

Answer (A). This question is discussed HERE.

Keep in mind that you don’t actually need to solve for an answer is DS; in fact, in some questions you will not be able to solve for the answer under the given time constraints. All you need to do is ensure that given unlimited time, you will get a unique answer.

Question 2: Four identical circles are drawn in a square such that each circle touches two sides of the square and two other circles (as shown in the figure below). If the side of the square is of length 20 cm, what is the area of the shaded region?

(A) 400 – 100π
(B) 200 – 50π
(C) 100 – 25π
(D) 8π
(E) 4π

Solution: First let’s recall that squares and circles are symmetrical figures. The given figure is symmetrical.

We don’t know any formula that will help us get the area of the curved shaded grey shape in the center. In such cases, very often what you need is to find the area of one region and subtract the area of another out of it. Here, if we subtract the area of the four circles out of the area of the square, the leftover area includes the shaded region but it includes other regions (around the corners etc) too. This is where symmetry helps us.

Notice that we can split the figure into four equal regions to get four smaller squares. Now focus on the diagram give below which shows you one such smaller square. The area around the four corners of the smaller squares is equal i.e. the area of the red region = area of the blue region = area of the yellow region = area of the green region.

Our shaded grey region has four such equal areas so

Area of the shaded grey region = Area of the smaller square – Area of one circle

Area of the shaded grey region = $(10)^2 – \pi(5)^2 = 100 – 25\pi$

Answer (C). This question is discussed HERE.

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INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 05:29

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Theory on Geometry:
Triangles
Polygons
Circles
Coordinate geometry
3-D Geometries

3-D Geometry Questions: https://gmatclub.com/forum/geometry-3-d ... 75006.html
Shaded Region Problems: https://gmatclub.com/forum/geometry-sha ... 75005.html
Probability and Geomtry: https://gmatclub.com/forum/special-quest ... l#p1374416

All DS Geometry to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=32
All PS Geometry to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=53

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

03 Jun 2015, 06:25

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PLEASE POST QUESTIONS/DOUBTS REGRADING THE QUESTIONS DISCUSSED ABOVE IN THEIR RESPECTIVE TOPICS.
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INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

13 Jul 2018, 22:07

Bunuel niks18 generis pushpitkc
KarishmaB chetan2u

My query is regarding: finding radius of a circle using side (a) of an equilateral triangle,
which is inscribed in a circle.

Since triangle is inscribed in a circle, all vertices of triangle must lie on circle.
The side of the equilateral triangle can not be diameter of the circle since, if I
use rule: angle subtended by the longest chord (ie diameter) one of angle of triangle
becomes 90 and stands invalid for equilateral triangle.

In theory explained in this post, did we purposely drew a perpendicular from A to BC ?
We can not have any estimate about angle BOC. Let me know if my understanding is correct
about how we formed 30-60-90 for triangle BOC.
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Bunuel

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

14 Jul 2018, 01:59

Kudos

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adkikani wrote:

Bunuel niks18 generis pushpitkc
KarishmaB chetan2u

My query is regarding: finding radius of a circle using side (a) of an equilateral triangle,
which is inscribed in a circle.

Since triangle is inscribed in a circle, all vertices of triangle must lie on circle.
The side of the equilateral triangle can not be diameter of the circle since, if I
use rule: angle subtended by the longest chord (ie diameter) one of angle of triangle
becomes 90 and stands invalid for equilateral triangle.

In theory explined on this post, did we purposely drew a perpendicular from A to BC ?
We can not have any estimate about angle BOC. Let me know if my understanding is correct
about how we formed 30-60-90 for triangle BOC.

From here: https://gmatclub.com/forum/math-triangles-87197.html

Equilateral triangle all sides have the same length.

• An equilateral triangle is also a regular polygon with all angles measuring 60°.
• The area is $A=a^2*\frac{\sqrt{3}}{4}$

• The perimeter is $P=3a$
• The radius of the circumscribed circle is $R=a*\frac{\sqrt{3}}{3}$
• The radius of the inscribed circle is $r=a*\frac{\sqrt{3}}{6}$
• And the altitude is $h=a*\frac{\sqrt{3}}{2}$ (Where $a$ is the length of a side.)
• For any point P within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.
• For a given perimeter equilateral triangle has the largest area.
• For a given area equilateral triangle has the smallest perimeter.
• With an equilateral triangle, the radius of the incircle is exactly half the radius of the circumcircle.
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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

13 Jul 2020, 02:11

Bunuel wrote:

Theory on Geometry:
Triangles
Polygons
Circles
Coordinate geometry
3-D Geometries

3-D Geometry Questions: https://gmatclub.com/forum/3-d-geometry- ... 71024.html
Shaded Region Problems: https://gmatclub.com/forum/new-geometry- ... 68426.html
Probability and Geomtry: https://gmatclub.com/forum/special-quest ... l#p1374416

All DS Geometry to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=32
All PS Geometry to practice: https://gmatclub.com/forum/search.php?se ... &tag_id=53

Hi Bunuel,

Wanted to let you know that first two links do not exist. Maybe you update those or remove?

Bunuel

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

13 Jul 2020, 02:17

Kudos

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minirana wrote:

Bunuel wrote:

Hi Bunuel,

Wanted to let you know that first two links do not exist. Maybe you update those or remove?

Fixed. Here are the links:

3-D Geometry Questions: https://gmatclub.com/forum/geometry-3-d ... 75006.html
Shaded Region Problems: https://gmatclub.com/forum/geometry-sha ... 75005.html
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INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

05 Sep 2021, 20:24

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Bunuel wrote:

CIRCLES AND INSCRIBED REGULAR POLYGON RELATIONS

By Karishma, Veritas Prep.

Attachment:

GeometryPost9Fig1.jpg

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GeometryPost9Fig31.jpg

How do we know that BO bisects ABD?

Bunuel

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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

06 Sep 2021, 01:37

Kudos

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shreyansh3 wrote:

Bunuel wrote:

CIRCLES AND INSCRIBED REGULAR POLYGON RELATIONS

By Karishma, Veritas Prep.

Attachment:

GeometryPost9Fig1.jpg

Attachment:

GeometryPost9Fig2.jpg

Attachment:

GeometryPost9Fig31.jpg

How do we know that BO bisects ABD?

Angles AOB, BOC, and AOC are 120 degrees each. Now, triangle BOC is isosceles (BO = CO = radius), ∠BOC = 120° and thus other two angles are equal and are 30° each. Same way you can derive that OBC = OBA = OAB = OAC = OCB = OCA = 30°.
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Re: INSCRIBED AND CIRCUMSCRIBED POLYGONS ON THE GMAT [#permalink]

08 Sep 2021, 05:24

Bunuel wrote:

shreyansh3 wrote:

Bunuel wrote:

CIRCLES AND INSCRIBED REGULAR POLYGON RELATIONS

By Karishma, Veritas Prep.

Thanks a lot! This cleared it

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