Math New Drill (Level 2) Geometry (Solid figure) Cut a big cube with 125 pieces of small cubes

There are white and black cubes whose one side is 1 cm. 

63 white cubes and 62 black cubes are arranged in by turns as shown in lower Fig. 1 and a large cube is made. 

Furthermore, this large cube is cut so that it may pass along three point A, B, and C. 

Answer the following questions. 

Noted that the volume of a triangular pyramid is calculated by the formula of (the area of the bottom) × (height) / 3. 

(1) Draw the situation of the cut face in the lower Fig. 2. 

(2) Find the ratio of the volume of a solid including the point D to the volume of the original large cube. 

(3) Find the number of white cube in a solid including the point D which is not cut. 

Math New Drill (Level 2) : Geometry (Solid figure) Problem 8 TOYOEIWA-2008

The vessel below is 10 cm in height and the bottom is the form which combined a rectangle and semicircles as shown in Fig. 1. 

Putting water into this vessel full and with attaching the portions of A and B to a floor, 45 degrees was leaned and water was spilt. 

Fig. 2 is a figure showing that it was seen from the side. 

Find the amount of the remaining water. 

Pi is assumed to be 3.14.

       

New Math Drill (Level 1) : Geometry(Solid figure) Problem3(1) NAKANO ATTACHED TO MEIJI UNV.-2014

There is a container whose all corners are the right angle as shown in the Fig.1 below. 

Water is put in 2/3 of the container. 

And the container is put on with the shaded face being placed in the bottom as shown in Fig.2. 

In this case, find the height of the water.

Math New Drill (Level 1) : Geometry(Solid figure) Problem2(3) NAKANO ATTACHED TO MEIJI UNV.-2014

10 pieces of dice are piled up on the desk so that the direction of each face might become same as shown in the figure below. 

Find the total of the number of dice spots not to be seen from anywhere.

Math New Drill (Level 2) : Geometry (Solid figure) Problem 10 OTSUMA-2014

The figure below is a solid which is made by hollowing out the cylinder whose diameter of the bottom is 2cm and height is 4cm from the cube whose length of one side is 4cm. 

Each center of the bottoms of the hollowed cylinder is the same as the diagonal of the intersection of square ABCD and square EFGH respectively. 

Point M is a middle point of the side CG.　

This solid is cut by the plane that passes three points E, F and M. 

Find the volume of the solid including face ABCD. 

Pi is set to be 3.14.

    

Answer

38.58 cm³

Solution

The volume of the solid ABCD-EFMN is 4 × 4 × 4 × 3/4 = 48 cm³. 

PS = 4 - ( 2 × 1/4) = 7/2 cm. 

QR = 4 - ( 2 × 3/4) = 5/2 cm. 

The volume of the below part of cylinder is 1 × 1 × 3.14 × (7/2 + 5/2) × 1/2 = 9.42 cm³ .

Therefore the volume to be found is 48 - 9.42 = 38.58 cm³.

   

Math New Drill (Level 2) : Geometry (Solid figure) Problem3 JOSAIKAWAGOE-2014

There are cubes of ① and ② as shown in a figure below. 

Cube ① is assembled with 8 pieces of small size cubes and cube ② is assembled with 27 pieces of small size cubes with the same small size. 

Answer to the following questions.

(1) When you cut cube ① by the plane which passes along the three points A, B, and C, answer the name of the figure of a cut surface as correctly as possible.

(2) When you cut cube ① by the plane which passes along the three points A, B, and C, answer the number of the small cubes cut.

(3) When you cut cube ② by the plane which passes along the three points A, B, and C, answer the number of the small cubes cut.

Math New Drill (Level 1) : Geometry (Solid figure) Problem1(7) JOSAIKAWAGOE-2014

Find the volume of the solid which is made by the rotation of the figure below around the line AB as a rotation axis.
Pi is assumed to be 3.14.

Math New Drill : SNJ00-0807-L2 Make rectangular prism with 8 pieces of rectangular prism

There are 8 rectangular prisms as shown in the figure below. 

All of them are used and those are piled up in the same direction so that a big rectangular prism is made. 
How many kinds of rectangular prism can be made?

New Math Drill : Level 2 : Geometry(Solid figure) Problem4 JISSENJOSHIGAKUEN-2014

Rotate the figure below around the straight line A as a rotation axis and make a solid. 

Pi is assumed to be 3.14.

(1) Find the area of the cut surface when you cut this solid by the plane including the axis.

(2) Find the volume of this solid.

New Math Drill : Level 1 : Geometry(Solid figure) Problem4 JOHOKUSAITAMA-2014

A cube whose length of one side is 6 cm is put on the flat floor as shown in the figure. 

This cube is rolled to the direction of the arrow with putting the side of FG on the floor so as not to be slipped until the face BFGC reaches on the floor. 

(1) Find the area of the part where the line segment AF moved. 

(2) Find the volume of the part where this cube moved.

Math New Drill : Level 1 : SNJ00-1105-L1 Incline cylinder containing water

There is a container of the shape of the column whose radius of the bottom is 8㎝, and height is 21㎝. 

This container is filled with water and it is inclined 45 degrees as shown in Fig.2. 

Find the volume of the water that remains without spilling in the container.

Math New Drill : SNJ00-1110-L2 Shortest line on trigonal pyramid

The figure below is a trigonal pyramid. △ABC is an equilateral triangle. OA = OB = OC. ∠ AOB = ∠ BOC = ∠COA 30 degrees. Point P, Q, R is fixed respectively on the side OB, OC, OA so that AP + PQ + QR + RB may be minimized. Find PQ : RB.

Math New Drill : Level 2 SNJ00-1116-L2 Draw water from water tank

There is an water tank shaped as a rectangular prism as shown in Fig.1 and the height of the surface of the water is 16 cm. 

There is a container shaped as a rectangular prism of 20 cm in depth as shown in Fig.2. 

The work with which water is drawn out of an water tank as much as possible using this container was done continuously twice. 

Answer the following questions. 

Noted that a triangle with which the ratio of length of three sides is 3 : 4 : 5 is the triangle with which the angle facing the longest side is 90 degrees. 

        

(1) Find the volume of the water which remaining in the water tank after drawn out for the 1st time. 

    (2) Find the volume of the water drawn out for the 2nd time.

Math New Drill : SNJ00-1115-L2 Triangular and quadrangular pyramid

There are some blocks of the quadrangular pyramid with 10 cm in length of all the sides and some blocks of the triangular pyramid with 10 cm in length of all the sides. 
(1) I make a triangular pyramid with 20 cm in length of all the sides by combining these blocks without any gap. 
How many blocks are used, respectively? 

(2) Find the volume ratio of one block of the quadrangular pyramid of one-side 10 cm and one block of the triangular pyramid of one-side 10 cm. 

Math New Drill : SNJ00-1113-L2 Rectangular prism made of 2 kinds of clay

There is a solid which two rectangular prisms made of the clays of black and white have stuck fast as shown in the figure below. 
Rectangular prism ABCD-EFGH is made of black clay. 
Rectangular prism DCJI-HGKL is made of white clay. 
This solid is cut by the plane which passes along three point A, F, and M. 
Find the ratio of the volume of black clay to the volume of white clay by the ratio of the least integer in the solid including the point C. 

Math Problem : KKK.22 Assembling development views of two solids

Fig.1 and Fig.2 show development views of solids.

The development view of Fig.1 consists of one rectangle, two equilateral triangles, and two trapezoids.

The development view of Fig.2 consists of four equilateral triangles.

Find the volume ratio of the solid made by assembling Fig.1 and the solid made by assembling Fig.2.

Math Problem : KKK.23 Image solid from development view

Find the volume of the solid which is made by assembling the development view of the figure.

Each triangular face is an equilateral triangle.

Each face of a hexagon is a made figure where one side cuts out two right-angled isosceles triangles whose equal lengths of two sides are 1 cm from the square which is 2 cm one side.

Math Problem : KKK.21 Vessel, water and bar

Water is contained in the vessel of a rectangular prism to a certain height as shown in Fig.1.
The bar of a rectangular prism whose length and width at the bottom is 10 cm and 6 cm respectively was put into this vessel to 24 cm from the bottom of the vessel for the both bottom might be parallel.
In this case, the water surface was up to the place of the height 9/10 of the vessel.

Furthermore, when the bar is put in until it reach to the bottom of the vessel, 600 cm³ of 

water was overflowed. 

Finally when the bar was taken out, the water surface came to 7/10 of the height of the vessel.



Answer the following questions.

(1) Find the ratio of the base area of the bar and the vessel.

(2) Find the depth of the vessel.

(3) Find the height of the water in the vessel first.

Math problem : KKK.20 Number of cut to make development view

Fig.1 is a solid of which surface consists of four equilateral triangles of the same size and this solid is called a regular tetrahedron.
Fig. 2 is a solid of which surface consists of six squares of the same size and this solid is called a cube.
Fig. 3 is a solid of which surface consists of eight equilateral triangles of the same size and this solid is called a regular octahedron.
Fig. 4 is a solid of which surface consists of 12 equilateral pentagons of the same size and this solid is called a regular dodecahedron.
Fig. 5 is a solid of which surface consists of 20 equilateral triangles of the same size and this solid is called a regular icosahedron.

It is considered that sides of these solid is cut by a cutter to be a development view.
Only the side is cut and a development view should not be separated into two or more sheets.

It is considered how many sides should be cut.

(Example)
In Fig. 1, if three sides are cut, it will become as it is shown in Fig. 6 or Fig. 7.

Since it will be separated into two sheets if four sides are cut as shown in Fig. 8, it does not suit conditions.
Therefore, the number of the sides to be cut is 3.
Find number of the sides which should be cut, respectively in the case of Fig. 2, Fig. 3, Fig. 4, and Fig. 5.

Math problem : KKK.19 Cut piling up eight cubes

As shown in a figure, eight cubes of 1 cm³ in volume are accumulated. 

When this solid is cut by the plane which passes along the side AB and CD, find the volume of the solid of the smaller one.