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How to conduct a lesson with entertaining tasks on the topic “Body volumes”?


The need to find the volumes of various bodies arose in antiquity. Today in the lesson we will find the volumes of various bodies using the previously obtained formulas. Some of the tasks may seem very easy to students, but it is with similar tasks that one has to meet in life, many of the tasks are not presented in your textbook. Problems similar to those discussed in the lesson may occur to you on the mathematics exam. Today we have an entertaining lesson.

In everyday life, we often come across bodies of different shapes and sizes. For example, we say that a bucket holds 10 liters of water. This means that the volume of the bucket is 10 liters. Another example: it took 15 cubic meters (or cubic meters) of wood to build a garden house. As you can see, in these examples, volumes are expressed in specific numbers, but in different units – in one case in liters, in the other – in cubic meters. In different units, the volume of the same body is expressed in different numbers. How, in practice, can you find the volume of this or that body? For example, how to measure the volume of a crocodile? How to find the volume of a camping tent? What formulas are used to calculate the volumes of bodies? The volume of a crocodile can be found by dipping it into a pool of water. To do this, you must first apply divisions on the walls of the pool, similar to those applied to the walls of the beaker. Then it should be noted: how much water was the original and what it became after the crocodile was lowered into the water. The difference in the numbers corresponding to these divisions will give us the volume of the crocodile. The described method of measuring volumes is not always convenient, if not at all applicable. If we want, for example, to find the volume of a railway tank car, then filling it with water and counting the number of liters of water is a rather tedious task. And for large structures, such as a large building or the Cheops pyramid, the method of measuring volumes by immersion in water is impossible. Moreover, in practice, usually even before the start of the construction of a structure, for example, the same railway tank car, you need to know what dimensions it should have in order to get the desired volume.

Lesson objectives:


  • expansion and deepening of students’ ideas about the volumes of various bodies,
  • ensuring the repetition, generalization and systematization of knowledge on finding the volumes of bodies.


  • contribute to the formation of skills to apply the techniques of generalization, comparison, transfer of knowledge to a new situation, the development of mathematical horizons, thinking and speech,
  • development of creativity, ability to analyze.


to contribute to the education of interest in mathematics, activity, the ability to communicate.


  • develop an interest in the subject,
  • continue the formation of general educational skills,
  • make a conclusion about the application of geometry in life,
  • develop different ways of analysis,
  • educate a mathematically literate person.

At the beginning of the lesson, it is necessary to repeat the formulas for the volumes of bodies: a cylinder, a cone, a ball, a straight prism.

Practical tasks

Problem 1

What part of the body volume is occupied by the water poured into it?
Answer: The cylinder is half filled with water.

Problem 2

What part of the body volume is occupied by the water poured into it?
The body is a cube. The unfilled part is a straight prism, the base of which is four times smaller than the face of the cube.
Answer: 0.75 body filled with water.

Problem 3

Standing on the ice, a boy weighing 45 kg throws a stone at an angle of 45 degrees to the horizon at a speed of 5 m / s.
Answer: the solution can be viewed in the application.

Problem 4

The radius of a watermelon is 11 times the thickness of its rind. What percentage of the weight of a watermelon is the rind?
Answer: 25%.

Problem 5

The round log weighs 30 kg. How much would a log weigh if it were three times thicker and half as short?
Answer: 4.5 times.

Problem 6

One person eats two chicken eggs for breakfast. An ostrich egg in all sizes (length, width, height) is 2.5 times larger than a chicken egg. How many people can I make a breakfast of 2 ostrich eggs for?
The volume of an ostrich egg is (2.5) 3 times larger than a hen’s egg. The volume is 15.625 times larger;
15.625: 2 7.81.
Answer: You can prepare breakfast for 7 people.

Problem 7

A person absorbs about 1.5 liters of liquid per day. For 70 years, this is 4000 buckets.

365 * 1.5 * 70 + 1.5 * 18 + 1.5 * 18 40000 l.

Bucket capacity 10L. (For comparison: the capacity of a railway tank car is 20,000 liters).
If the diameter of the bucket is not changed, but the buckets are placed on top of each other, then the height will change 4,000 times.
The height of a standard bucket is 27cm.
0.27 * 4000 = 1080m


Summing up, we can conclude that finding the volumes of bodies can come in handy in life. We all know many formulas. This knowledge can be useful to us throughout our lives. Geometry and our usual life are interconnected.

The problems considered may interest even students who are not fond of mathematics, they make it possible to look differently at the problems that mathematics poses for us.

As a homework assignment, students can be asked to solve the following problems:

The task is practical. Using the example of an orange, calculate what part of the whole orange is the peel (express the number as a percentage).
The task is similar to task # 1 in the lesson (but the water in the conical vessel reaches 1/4 of the height of the cone.
Given a conical vessel. (see problem 1). At what height will the water level be if the vessel is exactly half full.
Find problems in fiction related to the calculation of body volumes.
Two vessels are given. The first has the shape of a rectangular parallelepiped and its dimensions: 12; 12; … The second is the shape of a cylinder. At what height will the water be in a cylindrical vessel if water is poured into it from the first vessel. The radius of the base of the cylindrical vessel is 6.
Come up with a way to find the volume of a pear.
How many quail eggs should you use to make breakfast for 1 person. All sizes (length, width, height) of a quail egg are 3 times smaller than a chicken egg.
How far can a man in a boat see the sea? (take the height of the eyes of a seated person 0.6m). (Answer 3.6 km)
How high should a pilot climb to see himself around him at 50 km? (answer 200m)