Questions on Geometry: Geometric formulas answered by real tutors!

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use the formula for a trapezoid.

that formula is:

area =

b1 is the top base
b2 is the bottom base
a is the altitude (height) of the trapezoid.

attached reference helps you to see how this works.

http://www.mathopenref.com/trapezoidarea.html

how the formula was derived is a lot more complicated.

if you look at the picture of a trapezoid shown below, you will see that the trapezoid is composed of a rectangle and a triangle on each end.



The trapezoid in the picture is composed of the areas of 1+2+3.

2 is a rectangle.
1 and 3 are triangles.

if you complete a larger rectangle, then you will see that the area of this larger rectangle is composed of the areas of 1+2+3+4+5.

The area of 2 is made by the base b1 * the altitude (a).

The area of 1+2+3+4+5 is made by the base b2 * the altitude (a).

The formula for the area of the trapezoid is a*(b1+b2)/2 which is the same as:

(a*b1)/2 + (a*b2)/2

now a*b1 = the area called 2 which is the area of the smaller rectangle.

now also a*b2 = the area of (1+2+3+4+5) which is the area of the larger rectangle.

based on the previous 2 statements, the area of the trapezoid is equivalent to:

(2/2) + (1+2+3+4+5)/2

this is equivalent to:

the area of (2/2) + the area of (1/2) + (2/2) + (3/2) + (4/2) + (5/2)

now (2/2) + (2/2) = 2 so our formula becomes:

the area of 2 + the areas of (1/2) + (3/2) + (4/2) + (5/2)

we also know that the area of triangle 4 is equivalent to the area of triangle 1.

we also know that the area of triangle 5 is equivalent to the area of triangle 3.

This is because triangle 4 is congruent to triangle 1, and because triangle 5 is congruent to triangle 3. These triangles are congruent by SSS.

what this means is that we can substitute the area of triangle 1 for the area of triangle 4.

what this also means that we can substitute the area of triangle 3 for the area of triangle 5.

our formula of:

area of 2 + areas of (1/2) + (3/2) + (4/2) + (5/2)

can be made into:

area of 2 + areas of (1/2) + (1/2) + (3/2) + (3/2)

this is because 1 and 4 are equivalent, and 3 and 5 are equivalent.

since 1/2 + 1/2 = 1, and since 3/2 + 3/2 = 3, our formula becomes:

2 + 1 + 3.

if we reorder the terms, we get:

1 + 2 + 3 which is the area of our trapezoid.

by looking at each of the piece parts, we have determined that the area of the trapezoid given by the formula:

area = a*(b1+b2)/2

is equivalent to the formula:

area = (a*b1)/2 + (a*b2)/2 which is the area of the smaller rectangle divided by 2 plus the area of the larger rectangle divided by 2.

it's tricky, but it works.

if you can't deal with the complexity of the derivation, then just stick with the formula.

the formula for the area of a trapezoid is equal to (b1+b2)/2 * a.

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Probably the same way you would explain it to anyone else. The first thing you would need to do is to determine what the teacher listening to the explanation believes the word "trapezoid" to mean. If s/he is in the UK, s/he might believe the term "trapezoid" refers to a general quadrilateral having no parallel sides. In that case you would politely explain that there is no general method for finding the area of a trapezoid other than dividing the figure into two triangles, attempting to determine the areas of the two triangles, and then summing the two areas. On the other hand, particularly if your audience is from the US, I would say take the average of the measures of the two parallel sides, then multiply that result times the altitude of the trapezoid which is defined as the distance between the two parallel sides.

John

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area equals
where:
b1, b2 are the lengths of the two bases
a is the altitude of the trapezoid

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19.) Find the measures of an interior angle and an exterior angle of the indicated regular polygon.
Polygon: regular heptagon
A heptagon has 7 sides.
The sum of the exterior angles always totals 360 degs
Ext angles = 360/7 =~ 51.43 degrees.
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The interior angles are 180 - Ext angle = 180 - (360/7) = 900/7 degs
=~ 128.57 degs
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Then number 25 is:
Find the value of n for each regular n-gon described
N-gon: each interior angle of the regular n-gon has a measure of 165 degrees
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The easiest way to do these is to find the Exterior angle first, = 180-165
Ext = 15 degs
The sum of Ext angles = 360
n = 360/15 = 24 sides and angles.

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The height of a cylinder is six times its radius.
Find a function that models the volume V of the cylinder in terms of its radius r.
r = the radius,
6r = the height
:
V(r) =
:
V(r) =

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In a parallelogram adjacent angles are supplementary. From above we have
(i)
(ii)
we can solve for y in (ii) as
(iii)
then
(iv)
dividing, we get
(v)
Now we can solve for x in (i) as
(vi)
combine like terms to get
(vii)
subtract to get
(viii)
and divide to get
(ix)

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approach speed is 13 mph
32/13=2.5 hrs

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It's

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True
Special case square: Let each side = 1
each diagonal is sqrt(2) (Pythagoras)
2sqrt(2)=2.828 sum of the diagonals.
perimeter is 4
.
As one angle gets more acute and the other more obtuse the difference becomes even greater.
.
Ed

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???
It could be mislabeled 17 feet.

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Don't know, I can't see it.

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The side opposite the 30 degree angle is 1/2 of the hypotenuse or 4. The side opposite the 60 degree angle is the side opposite the 30 x sqrt(3) , or in our case 4sqrt(3).
Area = length x width = 4 x 4sqrt(3) = 16sqrt(3)

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describe each locus of point
1. in space, the locus of points equdistant from two parallel planes
Ans: A plane midway between the given planes.
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2. in a plane, the locus of points 4 cm from the center of a circle whose radius is 5cm
Ans: A cicle with center at the center of the given circle, with radius of 4 cm.
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3. in a plane, the locus of the centers of all circles with the radius 3 cm that are tangent to a given line L.
Ans: Two lines parallel to the tangent line and 3 cm from it.
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Cheers,
Stan H.

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insufficient data

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20*3 cm

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Your older sister is getting married and to save money she is making her own cylinder shaped cake with three layers. Top layer: Radius= 3cm Height = 3 cm Middle Layer: Radius= 6cm Height = 4 cm Bottom layer: Radius = 10cm Height= 5cm If each tub of frosting costs $6.99 and a each tub of frosting covers 120 cm^2 of cake, figure out how much money your sister will have to spend on frosting
.
Surface Area = Area of top +Area of the side
.
Surface Area = (Area of top) + (perimeter of top)* height
.
Surface Area = (pi r^2) + (2 pi r)* h
.
Top layer:
Surface Area = (3.14*3^2) + (2*3.14*3)* 3
Surface Area = (3.14*9) + (6*3.14)* 3
Surface Area = 84.78 cm^2
.
Middle layer:
Surface Area = (3.14*6^2) + (2*3.14*6)* 4
Surface Area = (3.14*36) + (12*3.14)* 4
Surface Area = 263.76 cm^2
.
Bottom layer:
Surface Area = (3.14*10^2) + (2*3.14*10)* 5
Surface Area = (3.14*100) + (20*3.14)* 5
Surface Area = 628 cm^2
.
Total area = 976.54 cm^2
.
Deciding how many tubs:
976.54/120 = 8.14 tubs
.
Since you can't buy a partial tub, you must buy the next whole number up:
9 tubs of frosting
.
Total cost:
9 * 6.99 = $62.91

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That is the wrong formula for surface area because we will never be covering the bottom of the cake and we also need to subtract the area of the layer above.
pi*r^2
3.14*6^2
pi*r^2
3.14*12^2
sa=2*pi*r*h+pi*r^2
sa=2*3.14*20*5+3.14*20^2
sa=2*3.14*12*4+3.14*12^2
sa=2*3.14*6*3+3.14*6^2
add those three and subtract
both
3.14*6^2
3.14*12^2
so we just add
sa=2*3.14*20*5+3.14*20^2
sa=2*3.14*12*4
sa=2*3.14*6*3

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It's the biggest one.

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That depends on how big "it" is.

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In 1 hour, from 0800 to 0900, Carlos has gone 50 miles.
Juanita gains on him at 10 mph (60-50).
She will pass him in 5 hours (50/10).
0900 + 500 = 1400, or 2 PM.
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I see no need for a diagram or a table.

?

The y-coordinate of the y-intercept of a plane is what you get 
when you substitute 0 for the other two letters and solve for y.






So the y-intercept is the point (x,y,z) = (0,4,0)

Edwin