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FOURTH GRADE / SOLID GEOMETRY 3.2 / 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 brick with dimensions 5 cm. , 9 cm. and 20 cm. weighs 900 g. Another one with different dimensions, 4 cm. , 15 cm. and 24 cm. , is of the same material, i.e. both have equal densities and different dimensions.
What is the weight of the 2nd brick ?

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



a) 1220 g.
b) 1539 g.
c) 1440 g.
d) 1280 g.


First find the volumes and their ratio (beter the bigger to the smaller ratio).
Do not forget : 900 x 1.6 = 90 x 16
y
2. The bigger brick of the previous question is cut between one of the corners and the half of one side as seen in the figure. What will be the weight of the remaining part seen in the figure ?

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

RY1 RY2



a) 1180 g.
b) 880 g.
c) 990 g.
d) 1080 g.
1/2 of 1/2, i.e. 1/4 is cut; 3/4 remains. Look at ry1 .
3. What is the name of the shape given below ?

!
a) triangular prism
b) prism
c) lateral prism
d) trapezoidal prism
Triangular, is it not? .
4. What is the volume of the triangular prism given below ?

!
a) 99.75 cm.3
b) 475/3 cm.3
c) 387/4 cm.3
d) 99 cm.3
First find the area of the equilateral triangle . Its base and height are 5 cm. and 4.3 cm. respectively. You need to multiply these and then divide by 2.
Then multiply the area by 9 which is the height of the prism (equilateral triangles will be treated as the bases).
5. The rectangular trapezoid and rectangles below are the net of the trapezoidal prism of question 2. What is the area of the figure, the whole figure ?

!


a) 586.6 cm.2
b) 556.4 cm.2
c) 466.4 cm.2
d) 716.8 cm.2
There are 4 rectangles and a trapezoid . Area of a rectangle = length x width
Area of a trapezoid = (Legth of bottom (15 cm.) + length of top (7.5 cm.)/2 x height (24 cm.) (in our case 22.5 x 12)
Since the 4 rectangles all have equal heights, find the sum of all lengths and multiply with this height (286.4 cm2) .
6. The figure below is the net of the triangular pyramid of question 4. What is the measure of its surface area ?

!


a) 155.4 cm.2
b) 156.5 cm.2
c) 216.6 cm.2
d) 216.8 cm.2
The height of the equilateral triangle is given. So are the sides of the rectangles .
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7. That the volume of a pyramid is 1/3 x base area x height is given in many books and references but it is not proven. Its real proof can be given only after a student learns calculus. A different geometrical proof is given in : ( Ref.- Volume of a Pyramid )
A possible way to understand the situation is considering slices of the pyramid. This may be considered as an introduction to calculus then and in the figure below are 3 slices considered.

!
What do we get for a formula for the volume of our right pyramid if we consider only these 3 slices ?

a) 1/3 x base area x height
b) 3/5 x base area x height
c) 1/8 x base area x height
d) 3/8 x base area x height
We get the average of 1, 3/4 and 1/2 for the remaining parts which ( the average) is 3/4 .
We have also to consider the factor 1/2 due to change of the shape (basic two dimensional form) from a rectangle to a right triangle .
8. Suppose that we have proven the volume of a pyramid is 1/3 x base area x height for our right pyramid . How can we generalize this to the pyramid shown below ?

!
a) it is a congruent figure
b) it consists of four pyramids of our type.
c) can not be done
d) it is a similar figure
The answer is easy to guess . We have not generalized to any pyramid yet, but we have to leave something for the future.
9. In question 7 we tried to find the formula for the volume of a pyramid and with our method we found the factor 3/8 which is 1/24 greater than the right factor. What may be the reason for this error ?

a) we have not considered the effect of surface slope of the plane ADCF
b) we have not considered all of the slices
c) our formula should be the right one
d) we have made a calculation error


3/8 - 1/3 = 9/24 - 8/24 = 1/24 .
10. In the figure left is our right pyramid and right its face ABF ignored net. What is the area of this one face ignored net ?

!


a) 234.25 cm.2
b) 84.75 cm.2
c) 184.75 cm.2
d) 154.45 cm.2
Do remember that the base is a rectangle .
5x9 + 5x7 + 9x7 + ... = 143 + ... = ...
11. How many cm.3 is the volume of the square pyramid of the figure below ?

!


a) 2306.7
b) 1823.5
c) 2527.4
d) 1694.5
(1/3) x 20 x 20 x 17.3 = ... .
12. Now we want to be sure of the formula and want to approximate the volume of the pyramid of previous question via a stepped pyramid. First we try 10 steps. The hegiht of each step (rectangular prism with a square base) has an height of 1.73 m.
Which result do we get for the volume of this stepped pyramid ?

!


a) 2464,2 cm.3
b) 2664,2 cm.3
c) 2188,8 cm.3
d) 1684,8 cm.3
Since the heights of all prisms are the same, it is enough to find the sum of the base areas of each rectangular prism and multiply this by the step height.
The base area of the lowest is 20 m. x 20 m. = 400 m2 . The next has 18 m. x 18 m. and so on.
Written in the reverse order is the sum 22 + 42 + 62 + ... + 182 + 202 = 1540
13. Having learned the method let us increase the number of steps to 50 and calcula- te the volume. Which choice should be our result ?
( The
sum of squares of even numbers from 2 to 100 is equal to 171 700. )

a) 2576.9 m.3
b) 2376.3 m.3
c) 1896.6 m.3
d) 1775.5 m.3
This time the height of each step is 17.3/50 = 0.346 m.
The top step has a square base of 20/50 = 2/5 m. side length. And by each step side length increases by 2/5 m.
14. What happens if we increse the number of steps to 100; what do we get as the volume ?

( The
sum of squares of natural numbers from 1 to 100 is equal to 338 350 )

a) 1541.8 m.3
b) 3241.4 m.3
c) 2341.4 m.3
d) 3341.8 m.3
This time the height of each step is 17.3/100 = 0.173 m.
The top step has a square base of 20/100 = 2/10 m. side length. And by each step side length increases by 2/10 m. .
15. The bar graph of percent relative error vs. number of steps is shown in the figure below. What is the percent relative error for the case of 100 steps ?

!


a) 15 %
b) 6 %
c) 3 %
d) 1.5 %
Percent relative error for 10 steps is (2664,2 - 2306.7)/2306.7 = 357.5/2306.7 = 0.15 i.e. 15 % .
16. In question 7 with only 3 slices we attained an accuracy of 4 % (1/24 = 4.2/100) which we could get with approximately 50 slices in question 12, 13, and 14. What may be the reason behind it ?

a) 3 slices are enough
b) by chance
c) There the change was linear (proportional with a variable, not with its powers)
d) it was a triangular figure
Add, if you can, two more slices just before and after the middle slice .
17. Which of the followings would be a better method to reduce percent relative error ?

!


a) to calculate the volume of the inner stepped (9 steps) pyramid and take the average of inner and outer
b) to reduce the number of slices
c) to arrange the slices for the best result
d) to apply different methods
look at the figure carefully .
18. How many cubes are there in the figure below ?

!


a) 15
b) 13
c) 11
d) 10
Look at the backs of cubes .
19. What may be the measures of the 0.33 liter aliminum can below ?

!


a) r=31.5 mm. and h=106 mm
b) r=41.5 mm. and h=106 mm
c) r=41.5 mm. and h=116 mm
d) r=51.5 mm. and h=96 mm
Calculate the volume for one of the given measurs and then try to guess .
20. How many nets of a cylinder may there be ?

!


a) 1
b) 2
c) 20
d) infinitely many
Remember, for the place of the circles there is no restriction except that they should be tangent to the sides or the rectange at opposite sides .
21. How many cm.2 is the surface area of the cylinder whose net is given below ?

!


a) 37500
b) 18700
c) 54500
d) 47100
Find the radiuses of the circles from the perimeter (long side length) of the rectangle .
Take π = 3.14
y
22. A cylinderical hole is drilled through a cube with the given measures. What is the volume of the remaining part ?

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

RY1 RY2



a) 0.6 x a2
b) 1.3 x a3
c) 0.8 x a3
d) 0.6 x a3


More than you guess . Take π = 3.2
23. The drilled cube of question 22 is cut by a plane as shown in the figure below. What will be the boundaries of intersection ?



a) a rectangle and an ellipse
b) a rectangle and a circle
c) a square and an ellipse
d) a square and a circle
It is not a vertical plane .
24. A cylinder of volume 750 cm.3 is cut by a plane passing through the center in a symmetrical way. What will be the volume of the lower part below the plane?



a) 175 cm.3
b) 375 cm.3
c) 275 cm.3
d) 325 cm.3
Sym = same metry = measure
25. What will be the greatest cross section a cylinder of radius r and height h may show ?

a) (1/2) x r x h
b) r x h
c) r x h3
d) 2 x r x h
The greatest possible intersection area with a plane ?
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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