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FOURTH GRADE / SOLID GEOMETRY 3.3 / SIMPLE BASIC SOLIDS

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

   SOLID SHAPES WE WILL INVESTIGATE ARE
 ♦ 
CUBES, PRISMS, PYRAMIDS, CONES and SPHERES
 ♦ THIS TIME WE WILL INVESTIGATE the CUBE AS THE BASIC SHAPE.

THERE IS ONLY ONE GOLDEN RULE:
1. EACH SHAPE HAS TO BE STUDIED SEPARATELY.
y
1. A ... is a three-dimensional geometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex .

!
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) cylinder
b) cone
c) sphere
d) pyramid


Cone is an important geometrical shape due to conics. Conics are the intersection of a cone and a plane. Look at ry1.
y
2. What do we get for the volume of the cone shown below if we do the same approximation as we did in question 12 of test-2 ?

!
In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) 2428.8 cm.3
b) 2417.8 cm.3
c) 1817.8 cm.3
d) 2628.8 cm.3
Have first a look at ry1 and try to calculate yourself.
Every cylinder has an height of 2 cm. The top cylinder's radius is 1 cm. and the radius increases by 1 cm. for each cylinder down the other.
So the volume is π x 2 x (1 + 2x2 + 3x3 + 4x4 + 5x5 + 6x6 + 7x7 + 8x8 + 9x9 + 10x10) cm.3
The sum squares of first ten natural numbers is 1+4+9+16+25+36+49+64+81+100 = 385 .
3. How many possible nets may a cone have ?

!
a) 1
b) many
c) 100
d) infinitely many
The base circle may at anywhere be tangent to the upper sector .
4. What should be the relation between h, r and α in order to have a cone net from the following figure ?

!
a) (360 - α)/360 = r
b) [(360 - α)/360] x h = r
c) [(360 - α)/360] x 2 x h = r
d) [(360 - α)/360] x h = 2 x r
The perimeter of the circular section of the the sector left should be as long as the circumference of the circle right.
5. What should be the measure of ∠A be for the following figure to be a cone net ?

!


a) 90 °
b) 108°
c) 180°
d) 80°
Remember the relation of the previous question .
6. What should be the surface area of the net in the figure if it is the net of a cone ?

!


a) 942 cm.2
b) 492 cm.2
c) 636 cm.2
d) 989 cm.2
π/2 x 400 + π x 100 = π x 300 .
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7. What will be the volume of the cone whose net is given in the figure below (the figure may be a litle bit distorted) if its height is 17.32 cm. ?

!
a) 1922.8 cm.3
b) 1812.8 cm.3
c) 1723 cm.3
d) 1818 cm.3
The measure of ∠A is 180° .
8. How many of the shapes in the figure below may be a cone net ?

!
a) 1
b) 2
c) 3
d) all
The orange circle is too big.
9. How much ice cream can the cone of the figure below take (π = 3) ?

!
a) 167 cm.3
b) 106 cm.3
c) 139 cm.3
d) 176 cm.3


Since π is taken to be 3 , π/3 = 1 . We only need to multiply square of the radius by height .
10. The light blue plane is parallel to the base of the cone. What will be the shape of the intersection curve ?

!


a) an ellipse
b) a circle
c) parabola
d) hyperbola
The conic curves (circle, ellipse, parabola, and hyperbola) are pretty important .
11. Which curve comes out as the intersection of the white grey-transparent plane and the cone ?

!


a) circle
b) parabola
c) an ellipse
d) hyperbola
As said, they should be learned .
12. How big is the volume of the lower red part if the plane is parallel to the base (π = 3) ?

!


a) 576 cm.3
b) 657 cm.3
c) 567 cm.3
d) 756 cm.3
It can be perceived as a fraction problem or as the difference of two cones.
As a fraction problem:
1- The upper cone's volume is 1/4 ths of the whole cone's if only the radiuses are considered (the volume is proportional with the square of the radius).
2- The upper cone's volume is one half of the whole cone's if only the heights are considered (the volume is proportional with height).
3- If both radiuses and heights are considered, the upper cone's volume is 1/8 ths of the whole cone's.
13. What is the volume of the following pyramid ?

!


a) 200 3
b) 400/3 cm.3
c) 133 3
d) 233 3
It is again one thirds of base area multiplied by height .
14. Could you give an example of a sphere that has its net on it, possibly found in your classroom ?

a) a spherical world atlas
b) a world atlas
c) a world map
d) any map
The sheet of paper covering the plastic sphere is supposed to be a sphere net .
15. What may the following figure be (Ref.- Q15) ?

!


a) a net
b) a figure
c) a sphere's net
d) an ink drop
This figure is called gores .
16. Which figure gives an idea about the shape of a sphere's net ?

!


a) II
b) I
c) both
d) none
Easy qustion .
17. What will be the volume of the sphere whose net is given below (π=3) ?

!


a) 5624 cm.3
b) 6524 cm.3
c) 5324 cm.3
d) 3524 cm.3
The radius of the sphere is given .
18. What will be the volume of the sphere shown in the figure below (π = 3) ?

!


a) 103 324 cm.3
b) 240 508 cm.3
c) 108 000 cm.3
d) 324 400 cm.3
6 times the arc length given will be the circumference from which the radius should be derived .
The radius will also be 30 cm. bacause π is given as 3 and 2 x π = 6 .
19. What may be the volume of the sphere embedded in the cylinder of the figure below (π = 3) ?

!


a) 62.5 cm.3
b) 53 cm.3
c) 53.25 cm.3
d) 24.5 cm.3
Look at the triangle carefully; it is the well known 3-4-5 triangle we have mentioned before (2.5 3 = 15.625) .
20. What may the volume of the cylinder that embeds the sphere be (π = 3) ?

!


a) 53.25 cm.3
b) 83.25 cm.3
c) 63.5 cm.3
d) 93.75 cm.3
Diameter is height .
21. Forget about the given data of the figure. What will be the volume ratios of the cylinder and the sphere if the second is embedded in the first as shown in the figure below ?

!


a) 3/2
b) 4/3
c) 2/3
d) 6/5
Compare 4/3 x π x (D/2)3 vs. π x (D/2)2 x D . Take the second to first ratio.
y
22. The red sphere has half of the radius of the grey sphere (shell). 1/8 ths of the grey spherical shell is cut and taken out. What will under these conditions be the ratio of the empty part to the full (solid) part ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) 1/7
b) 7/57
c) 17/29
d) 3


Let the radius of the red sphere be r . The volume of it will then be V = 4/3 x π x r3.For further help have please a look at ry1. Then, if you still have not found the result, look at the second y.
The volume of the solid part = 256/24 x π x r3 - 28/24 x π x r3 = 57/6 x π x r3.
The ratio is (28/24 = 7/6) ...
Another way to look at this problem : Let the volume of the red sphere be unit volume. The volume of the whole sphere without cut is 8 unit. The volume of the shell is 7 unit. Its cut part is 7/8 unit. Solid part is 8 - 7/8 = 57/8 unit. Since the denominators of 7/8 and 57/8 are both 8 , the ratio is ... .
23. What is the volume of the light blue part ?



a) 5/3 x π x r3
b) 2/5 x π x r3
c) 3/2 x π x r3
d) 2/3 x π x r3
The volume of the cylinder: π x r2 x 2 x r = 2 x π x r3 .
The volume of the sphere: 4/3 x π x r3
24. What is the volume ratio of the red part to the light blue part in the previous question ?

a) 2
b) 3/2
c) 3/4
d) 7/5
This is already solved indeed.
25. What is the ratio of surface areas of the cylinder and the sphere in the figure given in question 22 ?

a) 3/5
b) 2/3
c) 3/2
d) 5/3
The cylinders surface area is greater than of the sphere because of two reasons:
1. The volume of the cylinder is the greater volume.
Even if they had the same volumes the sphere is the compacter .
The ratio is supposed to be greater than 1 .
The surface area of the cylinder = 2 x π x r2 + 2 x π x r x 2 x r .
The surface area of the sphere = 4 x π x r2
This is the feedback!

THANK YOU FOR YOUR VISIT; HOPE WE COULD BE OF USE.
FOR MUTUAL BENEFIT YOU MAY USE OUR FORUM.
© 2011 , İsmail GERMAN. ALL RIGHTS RESERVED
PLEASE PREFER GIVING LINK IF THAT SUFFICES

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