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) MEIJI UNV.-NAKANO-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 Problem (Level 2) : Four solids in water tank (KK24)

In the water tank of the rectangular prism of Fig. 1, four solids as shown in Fig. 2 are placed for the shadow face to be as the bottom.

Then, water is poured into this water tank every 10 cm3/s.

(1) When the depth of water is to be 2 cm, find the area of the water surface.

(2) After (1), find the time when it will take until the water fills up the tank.

The volume of the solid of Fig. 2 can be calculated as (base area) × (height) × 1/3.

Math Problem (Level 2) : Three cylinders in water tank (KK23)

As shown in the figure, three cylinders A, B, and C were stood right in the water tank and water was put in until the height of the water surface was set to 18 cm. 

When A and B were taken out of the tank, the height of the water surface was 5 cm.

When A and C were taken out, the height of the water surface was 9 cm.

When B and C were taken out, the height of the water surface was 6 cm.

Answer the following questions.

(1) Find the ratio of the base area of the cylinders A, B, and C by the ratio of the least integer.

(2) Find the height of the water surface when only A is taken out.

Math Problem (Level 2) : Put two rectangular prism into water (KK22)

There are two large and small rectangular prisms 10 cm in height.

As shown in Fig. 1, a small rectangular prism is put on a large rectangular prism.

Moreover, water is contained in the tank of the rectangular prism as shown in Fig. 2.

The solid of Fig. 1 is straightly sunk to the bottom of the tank of Fig. 2 as it is.

When it is sunk when the initial depth of the water of a tank is 8 cm, the water surface will be gone up by 6 cm.

When it is sunk when the initial depth of water is 15 cm, the water surface will be gone up by 7.5 cm.

Find the base areas of two rectangular prism, respectively.

Math Problem (Level 2) : Cut line on the development view (KK20)

As for the cube in a figure, the points P and Q are middle points of the side AB and AD, respectively.

This cube was cut by the plane which passes along the three points P, Q, and G, and the side PQ was drawn as a part of the cut surface.

Draw other sides of the cut surface in a development view.

Math Problem (Level 2) : Difference of total surface area (KK19)

As shown in a figure, there is a cube whose length of one side is 5 cm and EP is parallel to BC.

The difference of the total surface area of two solids which are made by cutting this cube with the plane which passes along four point A, E, F, and D was 42 cm2.

In this case, find the length of BE.

Math Problem (Level 2) : Number of spots of dice (KK17)

The sum total of the spots of the upper face and down face of a die is set to 7. 

For example, the spots of the under face of the die of Fig. 1 is 6.

As shown in Fig. 2, the rectangular prism was made by attaching the same number of spots of six dice.

(1) Find the number of the spots of the face A.

(2) Find the sum total of the number of the spots of all the faces of the rectangular prism of Fig. 2.

Math Problem (Level 2) : Shortest line on solid (KK16)

There is a solid OABC surrounded with the four sheets of the equilateral triangle whose length of one side is 10 cm.

The point of the middle of the side AC is set to M and the points P and Q are on the sides OB and OC, respectively.

As shown in a figure, the straight line AP, PQ, and QM are drawn.

Find the ratio of AP : PQ : QM in case that the sum of the length of the straight line AP, PQ, and QM becomes the minimum.

Math Problem (Level 2) : Solid made by cutting quadrangular pyramid (KK13)

The figure is a development view of the solid which was made of cutting a certain quadrangular pyramid by the plane parallel to the bottom.

(1) Find the height of the original quadrangular pyramid.

(2) Find for the volume of this solid.

Math Problem (level 2) : Projection view of solid (KK12)

When a certain solid surrounded in the five faces is seen from right above, it is as it is shown in Fig. 1, and when it is seen from right in front, it is as it is shown in Fig. 2. 

Find the volume of this solid. 

Math Problem (Level 2) : Development view of rectangular prism (KK11)

The character P is written to one face of a certain rectangular prism. 

Moreover, one diagonal line is drawn on each face of three faces. 

Fig. 1 and Fig. 2 shows two types of the development view of this rectangular prism. 

The length of the side AB is 14 cm.

(1) Draw three diagonal lines on Fig. 2 in handwriting.

(2) It is 24 cm when the length of the circumference of rectangle ABCD is subtracted from the length of the circumference (bold line) of the developed view of Fig. 1. 

Moreover, it is 10 cm when the length of the circumference of the developed view of Fig. 1 is subtracted from the length of the circumference (bold line) of the development view of Fig. 2. 

Find the volume of this rectangular prism. 

Math Problem (Level 2) : Assemble development view (KK10)

As shown in the figure below, the development view was drawn in the grid sheet 1 cm in one scale.

Find the total surface area and the volume of the solid made of being assembled this development view. 

Math Problem (Level 2) : Triangular pyramid from rectangular paper (KK8)

Some rectangular pasteboard is cut off along a diagonal line, as shown in Fig. 1. 

By using this I would like to make the glass which has the form of the triangular pyramid as shown in Fig. 2.

However, it may be unable to make or able to make depending on the length of the side AB.

Answer the conditions of the length of the side AB in the case of being able to make. 

Math Problem (Level 2) : Cut cube to be development view (KK7)

If the cube is cut along with the bold line of the figure, which is the right development view?

Choose the right one from ① to ⑧ and answer the number.

Both of front and back sides shall not be distinguished.

Math Problem (Level 2) : Shadow of quadratic prism (KK4)

There is a big wall which stands vertically to the ground.

The bottom of the square pillar of Fig.2 is a trapezoid as shown in Fig.1.

The height is 30 cm. 

It is placed as the side CD is parallel to the wall and the distance to the wall is 10 cm.

The angle to look up at the sun from the ground is 45 degrees.

The shadow area of Fig.2 expresses the shadow made on the ground of this quadratic prism.

Find the area of the whole shadow made to the wall by this quadratic prism.

Math Problem (Level 2) : Development view of quadrangular pyramid (KK3)

The shadow portion in the figure is made of cutting off four isosceles triangles 2cm in height from the square which is 8 cm one side.

Find the volume of the quadrangular pyramid made by assembling this development.