Questions on Geometry: Polygons answered by real tutors!

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A cube has an angle length of 4 inches. The cube is painted red all over and then cut into 64 cubes of edge length 1 inch. How many of these cubes have one face painted red?
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I don't know what "angle length" is.
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If you meant the length of a side:
On each face, the cubes will have more than one side painted. Each face is cut into squares 4 by 4, so the inside squares (2 by 2) will be the cubes with only one side painted. There are 4 on each of the 6 faces, so
6*4 = 24

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A dissection of a rectangular hexagon into 8 congruent quadrilaterals?
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I'd like to see that, too. A rectangular hexagon would be interesting by itself.

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A triangle doesn't have a radius.

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Please help me with this problem! Find the number of the sides of a regular polygon if each exterior angle measure is 18?? I'm stumped.
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The sum of the INTERIOR angle of a polygon with n sides is 180*(n-2). Each angle is (180*(n-2))/n.
The exterior angles are 180-interior angle, so this is a polygon with interior angles of 162 degrees.
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So, 180*(n-2)/n = 162
180*(n-2) = 162n
180n-360 = 162n
18n = 360
n = 20
I don't remember the name of that one, maybe duodecagon.

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An equiangular (equi - equal and angular - angle) polygon is a polygon whose angles all have the same measure

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Quadrilateral ABCD is made of the line segments AB, BC, CD, and DA (or AD)


Quadrilateral UVWX is made of the line segments UV, VW, WX, and XU (or UX)


So the corresponding parts are

AB <---> UV
BC <---> VW
CD <---> WX
DA <---> XU


Since quadrilateral ABCD is congruent to quadrilateral UVWX, this means that AB is congruent to UV, BC is congruent to VW, CD is congruent to WX, and DA is congruent to XU


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Answer:

So BC is congruent to VW

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Can you be more specific with what you mean by dilation?
There are a couple of different uses of that term in algebra.
Since you posted under geometry perhaps that's what you meant.
Here are two webpages with examples that do an excellent job of explaining.
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http://regentsprep.org/regents/math/dilate/Ldilate.htm
http://regentsprep.org/regents/math/codilate/ldilate.htm
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If that doesn't satisfy you or isn't what you meant, please re-post.

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circumference of a circle:
2(pi)r
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where pi is 3.14
.
30.2 = 2(3.14)r
15.1 = 3.14r
4.81 cm = r (radius)

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Area of any circle is:
(pi)r^2
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They gave you the area as 547 and pi is 3.14. Plug and solve:
547 = (pi)r^2
547 = (3.14)r^2
174.2 = r^2
13.2 m = r (radius)

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A Regular polygon is a polygon with all sides are all the same length and the angles are all equal.( ex. equilateral triangle, square, regular octagon, etc)
see www.mathleauge.com
Thank you,
Jojo

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As triangle PRQ is an isosceles triangle with PQ = RQ,
< RPQ = < PRQ = 32°
So in triangle PRQ, < Q = 180° - < RPQ - < PRQ = 180° -32° - 32° = 116°

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One of the base angles of an isosceles triangle has a measure of 65. What is the measure of the vertex angle?
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The 2 base angles are equal, so the other is also 65º. They add to 130º. The sum of the angles of all triangles is 180, so the vertex angle is 50º.

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In triangle DCS, the measure of angle D=58, the measure of angle C=80, and the measure of angle S=42. Name its sides in the order of incresing lengths.
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S, D, C. The sides are usually designated by the same letter as the opposite angle. Usually lower case is used for the sides, tho. s, d, c.

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The sum of the angles in a triangle is 180º. The 3rd angle has to be 180-100, or 80º.

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9^2+14^2=x^2
81+196=x^2
x^2=277
x=sqrt277
x=16.6433 answer.
9^2+x^2=14^2
81+x^2=196
x^2=196-81
x^2=115
x=sqrt115
x=10.7238 answer.

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47+x+90=180
x=180-90-47
x=43 degrees is the missing angle.

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Ok, here's the plan (strategy)!
We need to find the area of each of the four congruent isosceles triangular faces. To do this we need to find the slant-height of each face, so let's do this first:
The slant-height is represented by the hypotenuse of the right triangle formed by the height of the pyramid (220 m.) and the base which is exactly half the length of one side of the square base of the pyramid (48 m.).
We'll use the Pythagorean theorem to find the length of the slant-height (hypotenuse)" where:
c = the slant-height.
a = the length of the base of the right triangle (48 m.).
b = the height of the pyramid (220 m.).



Approx.
Now we can find the area of one of the buliding's triangular faces.
The base of the triangle here would be the length of one side of the square base of the building (96 m.), and the height of course is the slant-height just calculated (225 m.).


sq. meters.
To find the area of the four faces of the building, multiply by 4.

sq.meters.
Now for each 250 sq.meters of building surface, we need 1 gallon of cleaning solution, so we'll divide the toal surface area (43200 sq.m.) by 250.
Rounding to the nearest gallon, we get 173 gallons of cleaning solution.

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What is the size of each interior angle of a regular polygon if the number of sides is 4? 5? 6? 7? 8? 9?
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The number of sides is the same as the number of angles and vertices.
The total of the interior angles of a polygon is 180*(n-2), where n is the number of sides, angles and vertices.
For regular polygons, all angles are equal, so they're the total/n.
That is 180*(n-2)/n
Just sub the number for n:
For 4, it's 360/4 = 90 degs
For 5, it's 540/5 = 108 degs
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For 9, it's 1260/9 = 140 degs
For 36, it's 6120/36 = 170 degs
Notice that the angle increase as the number of sides increase. You can use that as a check for polygons with 5, 6, 7 & 8 sides.
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For 1000, it's 179.64 degs
As n approaches infinity (which is a circle), the angle approaches 180 degs.

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S=n(180-(360/n)) __ S=180n(1-(2/n)) __ S=180(n-2) __ S=180n-360

S2=180(2n-2) __ S2=2(180n-180) __ S2=2(180n-360+180) __ substituting __ S2=2(S+180)

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found the answer on the web.
take the hexagon and draw diagonals through each of the opposite corners.
this breaks the hexagon up into 6 equilateral triangles.
each of the sides of these equilateral triangles will be 12 inches long.
area of a triangle is 1/2 base * height.
we have the base.
we need the height.
take one of the equilateral triangles and drop a perpendicular from opposite side.
this becomes the height of the triangle.
the right triangles become 30-60-90 triangles.
in a 30-60-90 triangle, the sides opposite the angles are in the following ratio:
side opposite 30 degrees = .5
side opposite 60 degrees =
side opposite 90 degrees = 1
height is side opposite 60 degrees.
base is side opposite 30 degrees.
since the base is (1/2)*L and the hypotenuse equals L, the height is equal to
*L
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we now have enough to compute the area of the hexagon.
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area of the hexagon is 6 * the area of each equilateral triangle.
area of each equilateral triangle is 1/2 * base * height.
base = 12
height = *12
formula becomes
6 * (1/2) * 12 * *12
answer is:
374.1229744.
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formula for area of the hexagon had become

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An 8 sided polygon is an octagon with diagonal sides.
Remember: Probability=no. of ways it'll occur/total no. of possible outcome ------>1
So if Anu chose 1 diagonal, the probability Anitha will land on that diag. is:
or 25% ----> see 1 above.---------> FINAL ANSWER
If Anu chose 2 sides, then the probability that Anitha will land on those sides chosen by Anu: or 50%. ----> see 1 above.
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Now if it's SOLID Octagon, it's different:

see www.worselyshool.net/science/files/polygonintro
Then: = =5%, ANSWER
Thank you,
Jojo

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I'm assuming that you mean a "regular" polygon!
The measure of an interior angle, A, of a regular polygon of N sides is given by:
Substitute A = 120 degrees.
Multiply both sides by N.
Subtract 120N from both sides.
Add 360 to both sides.
Finally, divide both sides by 60.

The regular polygon has 6 equal sides and is called a hexagon.

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A regular hexagon has six sides of equal length.
So its perimeter is 6*(3x + 5) = 18x + 30.

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Rhombus
I have all of the properties of the parallelogram PLUS
- 4 congruent sides
- diagonals bisect angles
- diagonals perpendicular
The answer is a rhombus.

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Nobody would have coined a special name for that.
They'd just use the rule 

"An n-gon is a polygon with n sides".

So it would just be called a "9999-gon".

Edwin

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The sum of the measures of the interior angles of a regular polygon is:
180(n-2)
Setting it equal to 540, we have
180(n-2)=540
Solving for n, we obtain
180n-360=540
180n=900
n=5
So the polygon has 5 sides.

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( n - 2 ) * 180 = 540

n - 2 = 540 ÷ 180

n - 2 = 3
n = 2 + 3
n = 5
then its called a " Penagon "

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The formula for the sum of interior angles for an n-sided polygon is

In your case,



A five sided polygon is a pentagon.

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To solve this problem, you need to know the volume of each pillar. The volume is "area of the base" times "height".
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"area of the base" = "area of an octagon"
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Without trying to derive the formula of an octagon, I'll just use it. But, for reference, you might want to check out:
http://www.mathreference.com/geo,hex.html
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"area of an octagon" = 2(1+sqrt(2))S^2
where S=length of one edge
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So, plugging in your numbers to:
"area of an octagon" = 2(1+sqrt(2))S^2
"area of an octagon" = 2(1+sqrt(2))6.5^2
"area of an octagon" = 2(1+sqrt(2))42.45
"area of an octagon" = 2(1+1.414)42.45
"area of an octagon" = 2(2.414)42.45
"area of an octagon" = (4.828)42.45
"area of an octagon" = 204 sq ft
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Now, to get the volume we multiply the area times the height:
"vol of one pillar" = "area of an octagon" * "height of pillar"
"vol of one pillar" = 204 * 9
"vol of one pillar" = 1836 cubic feet
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Since we have 2 pillars, we multiply the above by 2:
2 * 1836 = 3672 cubic feet
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That's how much granite you need.

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A kite has an 8 inch side and a 15 inch side which form a right angle. find the length of the diagonals of the kite.
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If a triangle has a "right angle", then you can apply Pythagorean theorem:
Let x = diagonal of kite
then
x^2 = 8^2 + 15^2
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Now, solving for 'x':
x^2 = 8^2 + 15^2
x^2 = 64 + 225
x^2 = 289
x = 17 inches

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Let's name the parallel sides as and
And the non parallel sides as "c=?" and "d=?".
Before I proceed farther, I suggest you draw it on your own while we discuss it by words so you can follow better (just a suggestion).
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Now from that starting point and also the endpoint of that 9 inches which is , draw a line all the way down to which is perpendicular right? --> mark these lines as "h" (for height),likewise the 2 lines are parallel, and that line inside these parallel lines on also measures 9 inches right (of course) ---> mark it
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Now you have remaining length of 3 inches,
And these 3 inches will be cut into half, 1-1/2" on one side, likewise 1-1/2" on the other. Why? Because you have opposite similar 65 deg angle right? (If not the same angle, the length won't be the same)
The 1-1/2" length mark one side as and the other as
.
Can you follow still?
Now, side "c", "h" and forms a right triangle (if you mark them properly) with angle 65 deg. To get "c",

= 1.5/cos65
c=3.55 inches = d -------------> length of the non parallel sides
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For area in the trapezoid,
A=1/2(h)
To get "h",
h=(sin65)(3.55)=3.22 inches
Therefore,
A=(1/2)(3.22){9+12}
A=33.81 sq inches
Thank you,
Jojo

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a hexagon is inscribed in a circle of radius 4. find the area of the hexagon.
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Assuming it's a regular hexagon, there are 6 equilateral triangles with sides of 4.
The area of each is bh/2, =
The total area is 6 times that,
=
= apx 41.57 square units
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As a check, the area of the circle is = 50.265
The circle will always be more than anything inscribed in it.

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Pentagon=540 degrees
540/5=108
108=the vertex angle of the isosceles triangle

1/2 of 108=54
tan(54)=x/10
x=13.764
(x=length of half of one side of the pentagon)
2x=27.528

apothem=10
perimeter=137.638

area=1/2ap

137.638(10)=1376.38

1376.38/2=688.19

area=688.19

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A dedecagon(sic)?...I don't think there is any such thing!
Perhaps you meant "dodecagon" which is a 12-sided polygon.

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1. Two angles are supplementary, and the ratio of their measures is 7 to 2 what are the angle measures?
When two angles are supplementary, the sum of the two angles equals 90.
7/9 * 90 = 70 deg
2/9 * 90 = 20 deg
2. Two angles are complementary, and the ratio of their measures is 5 to 4. What are the angle measures?
When two angles are complementary, the sum of the two angles equals 180.
5/9 * 180 = 100 deg
4/9 * 180 = 80 deg

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In a regular polygon of n sides, the measure of each of the (equal) interior angles is given by:
Substitute and solve for n.
Multiply both sides by n and simplify.
Add 360 to both sides and subtract 144n from both sides.
Divide both sides by 36.
A decagon.

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First in since the triangle is an equilateral triangle, all sides are equal, and thus each side is 6. Also that means that all the angles of the triangle are congruent, and in since a triangle must have 180 degrees, we divide 180 by 3 and get 60. Thus each angle is 60.
Then we use the area formula.
K= .5 AB Sine C
K = .5 6(6) Sine 60
K= 15.5

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Multiply both sides by 13.

This is reduced.

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You can use Heron's formula for finding the area of any triangle when you know only the lengths of the three sides.
Heron's formula is:
where s = the semi-perimeter of the triangle and a, b, and c are the lengths of the three sides.
In your problem, = and a, b, and c = 6 inches each.




sq.in. to the nearest tenth.

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A regular tetrahedron is composed of 4 triangular faces, here all sides are 12"
The area of a triangle is 1/2 of base times height, bh/2.
The height is
h =
The area of 1 triangle is
The total is 4 times this, or
Area = 249.415 square cm

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You haven't indicated whether or not this is a REGULAR polygon (one whose sides are all of equal length) but I'll assume that it is.
You can start with the formula for the sum of the interior angles (S) of a regular polygon of n-sides:
Since you know that this sum is 1440 degrees, you can substitute this into the formula and solve for n, the number of sides.
Divide both sides by 180
Add 2 to both sides.

The regular polygon has 10 sides. This is known as a decagon.

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Given square ABCD with AB= 2x+1 and CD= 5x-19 what is the perimeter of the square?
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All the sides are equal so:
2x+1 = 5x-19
3x = 20
x = 20/3
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One of the sides is 2x+1 = 2(20/3)+1 = 43/3
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The perimeter = 4(43/3) = 172/3 = 57 1/3
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Cheers,
Stan H.

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The sum of the interior angles is (n-2)180
EQUATION:
(n-2)180 = 1080
n-2 = 6
n = 8
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the figure is an octagon
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Cheers,
Stan H.