Through the midline of the base of a triangular prism is the volume. Through the midline of the base of a triangular prism

Hello! Another portion of problems with prisms, triangular prisms are considered. I combined several tasks similar in one “attribute” - they have, through midline the base passes through the section. The questions are about calculating the surface area or volume of either the original prism or the cut one. What is important to remember here?

This property of similarity of figures relating to area, in particular about the triangle, was already discussed in one of the articles. But even if you suddenly forget this, the presented tasks will be intuitive and you will solve them in one action.

77111. Through the center line of the base triangular prism, the lateral surface area of which is 6, a plane is drawn parallel to the lateral edge. Find the lateral surface area of the trimmed triangular prism.

It is said that the plane passes through the midline of the base, that is, through the points that are the midpoints of the adjacent sides of the triangle. Moreover, it runs parallel to the side edge - this means that the specified plane also passes through the midpoints of the corresponding adjacent sides of the other base.

Without any calculations, it is clear that the lateral surface area of the cut-off prism will be half that of the original one.

Look!

The prisms have a common height. The specified plane cuts two adjacent side faces in half.

Let's consider the third face ( parallel to the plane section) - its surface area is also half as large, since the middle line of the triangle is half the size of the side of the triangle parallel to it.

Considering that the height remains unchanged (common for both prisms), we can conclude that the lateral surface area (the sum of the areas of all three faces) of the cut-off prism will be half as large.

Answer: 3

76147. A plane parallel to the side edge is drawn through the center line of the base of a triangular prism. The lateral surface area of the trimmed triangular prism is 20. Find the lateral surface area of the original prism.

The task is the opposite of the previous one. FFormula for the lateral surface area of a prism:

This means for a cut-off prism:

The height of the pyramids is common, so the area of the lateral surface of the original prism depends on the perimeter. Since the resulting triangles at the base of the prism are similar, and their corresponding sides are in the ratio 1:2, this means that the perimeter of the base of the original prism is twice the perimeter of the base of the cut-off one.

This means that the lateral surface area is also 2 times larger and equal to 40.

Answer: 40

27106. Through the middle line of the base of a triangular prism, the volume of which is 32, a plane is drawn parallel to the side edge. Find the volume of the cut-off triangular prism.

It is known that the volume of the prism equal to the product base area and height. The height for these prisms is common, which means the change in volume depends only on the change in surface area.

Let's consider the triangles lying at the bases of the prisms - they are similar. If we consider the base of the original prism relative to the base of the cut-off one, then the similarity coefficient will be equal to 2. What does this give us?

We know that have mercy similar figures correlate as the square of the similarity coefficient, which means:

The base of the cut-off prism is 4 times smaller.

Thus, its volume will be 4 times smaller, that is, 8.

Formally, it can be written like this:

Answer: 8

74745. A plane parallel to the side edge is drawn through the center line of the base of a triangular prism. The volume of the cut-off triangular prism is 7. Find the volume of the original prism.

The task is the opposite of the previous one. The volume of the prism is equal to the product of the area of the base and the height:

The height is total, which means the volume changes depending on the change in the area of the base.

The triangle lying at the base of the original prism, as already said, is similar to the triangle lying at the base of the cut-off prism. The similarity coefficient is 2, since the section is drawn through the center line.

The areas of similar figures are related as the square of the similarity coefficient, that is:

Thus, the base area of the original prism more area base of the cut-off prism 4 times.

Therefore, the volume of the original prism will be 4 times greater than the volume of the cut-off prism. Thus, the required volume is 28.

Answer: 28

Three more problems about the surface area of a prism

245356. The surface area of a regular triangular prism is 6. What will be the surface area of the prism if all its edges are tripled?

Let's enlarge all the edges of the prism three times. What happens?

It turns out that each face of the resulting prism and the corresponding face of the original prism are similar figures. Moreover, the similarity coefficient is equal to 3. We know that the areas of similar figures are proportional to the square of the similarity coefficient, that is:

This means that the area of each individual face of our prism will increase by 9 times. Since the surface area of the entire prism is the sum of the areas of all the faces, it goes without saying that the entire surface area of the prism will also increase by 9 times.

Answer: 54

*In fact, it doesn’t matter what kind of body we are talking about (a prism, a pyramid, a cube, a parallelepiped), the essence is the same.

In a triangular prism, the two side faces are perpendicular. Their common edge is 30 and is separated from the other lateral edges by 3 and 4. Find the lateral surface area of this prism.

At the time of writing, this task is from open bank The Unified State Exam assignments have been deleted, but we will look at it, so you can return there at any time, which means you can take the exam in future years.

To calculate the lateral surface of the prism, we use the formula:

IN in this case side rib this is the common edge of the faces perpendicular to each other, it is equal to 30. The perpendicular section of the prism is right triangle with legs 3 and 4. Using the Pythagorean theorem, we find its hypotenuse and can calculate the perimeter:

Thus:

Second solution!

The above formula may not be clear to some. What is its meaning and what does it express?

Look at each individual face (with the prism on its side) - these are parallelograms. Moreover, the bases of these parallelograms are equal and they are equal to the side edge, that is, 30. Their heights will be different.

Two are known to us 3 and 4, the third is unknown. But we can find it. Let's cut the prism perpendicular to the side edges, the section of the cut will be a right triangle with legs 3 and 4, find the hypotenuse:

It turns out that the lateral surface area is equal to the sum three squares parallelograms:

Answer: 360

72605. Find the lateral surface area of the regular hexagonal prism, the side of the base is 6 and the height is 2.

Solution.

Answer: 12.

Answer: 12

The area of the lateral surface of a triangular prism is 6. A plane parallel to the side edge is drawn through the middle line of the base of the prism. Find the area of the lateral surface of the cut-off triangular prism.

Solution.

prototype.

The area of the side faces of the cut-off prism is half the corresponding areas of the side faces of the original prism. Therefore, the area of the lateral surface of the cut-off prism is doubled less area lateral surface of the original one.

Answer: 12.

Answer: 3

The area of the lateral surface of a triangular prism is 26. A plane parallel to the side edge is drawn through the middle line of the base of the prism. Find the area of the lateral surface of the cut-off triangular prism.

Solution.

This task has not yet been solved, we present the prototype solution.

The area of the lateral surface of a triangular prism is 24. A plane parallel to the side edge is drawn through the middle line of the base of the prism. Find the lateral surface area of the cut-off triangular prism.