Calculation of the volume of figures. How to calculate the volume of an irregularly shaped body. Volumetric geometric shapes

– a free geometric calculator will help you calculate the area or volume of relatively simple geometric shapes. No need to search for the necessary formulas and make calculations on a piece of paper. Working with the program is very simple, first you need to choose what you need to calculate: the area of \u200b\u200bthe figure, the area full surface or the volume. The selected figure is displayed next to the window, and next to it will be shown a formula for calculating the desired value. Initially, all results are rounded to the integer part, but it is possible to change and select the required accuracy with which the results should be displayed. For this, options are available from one to ten decimal places.

What can be calculated?

Circle - find the circumference of the known radius, and the diameter of the known circle.
We find the area - a circle, a sector of a circle, an ellipse, a square, a rectangle, a parallelogram, a triangle, a trapezoid, a rhombus, a torus.
Surface area - cube, prism, pyramid, cylinder, sphere, cone, torus.
The volume of figures - cube, cuboid, prism, pyramid, cylinder, spheres, cones, torus, frustum, barrel.

Make sure the body is waterproof, as the method described involves immersing the body in water. If the body is hollow or water can penetrate it, then you will not be able to accurately determine its volume using this method. If the body absorbs water, make sure the water will not damage it. Do not immerse electrical or electronic items in water as this may cause injury. electric shock and/or damage to the item itself.

If possible, seal the body in a waterproof plastic bag (after releasing the air). In this case, you will calculate a fairly accurate value for the volume of the body, since the volume of the plastic bag is likely to be small (compared to the volume of the body).

Find a container that holds the body whose volume you are calculating. If you are measuring the volume of a small object, use a measuring cup with a graduation (scale) of volume. Otherwise, find a container whose volume can be easily calculated, such as a cuboid, cube, or cylinder (a glass can also be thought of as a cylindrical container).

Take a dry towel to lay the body out of the water on.

Fill the container with water so that the body can be completely immersed in it, but at the same time leave enough space between the surface of the water and the top edge of the container. If the base of the body has an irregular shape, such as rounded bottom corners, fill the tank so that the water surface reaches a part of the body that is regularly shaped, such as straight rectangular walls.

Note the water level. If the water container is transparent, mark the level on the outside of the container with a waterproof marker. Otherwise, mark the water level on the inside of the container using colored tape.

Immerse your body completely in water. If it absorbs water, wait at least thirty seconds and then pull the body out of the water. The water level must go down because some of the water is in the body. Remove marks (marker or adhesive tape) from the previous water level and mark the new level. Then once again immerse the body in water and leave it there.

If the body is floating, attach a heavy object to it (as a sinker) and continue the calculation with it. After that, repeat the calculation exclusively with the sinker to find its volume. Then subtract the volume of the lead from the volume of the body with the weight attached and you will find the volume of the body.

When calculating the volume of the sinker, attach to it what you used to attach the sinker to the body in question (for example, tape or pins).

Mark the water level with the body submerged in it. If you are using a measuring cup, record the water level according to the scale on the cup. Now you can pull the body out of the water. It is probably not a good idea to leave the item under water for more than a couple of minutes, as the water may otherwise adversely affect it.

Know why this method works. The change in the volume of water is equal to the volume of the body irregular shape. The method for measuring the volume of a body using a container of water is based on the fact that when a body is immersed in a liquid, the volume of the liquid with the body immersed in it increases by the volume of the body (that is, the body displaces a volume of water equal to the volume of this body). Depending on the shape of the water container used, there are various ways calculating the volume of water displaced, which is equal to the volume of the body.

Find the volume using the beaker's measuring scale. If you used a container with a measuring scale, then you should already have two values \u200b\u200bof the water level (its volume) recorded. In this case, from the value of the volume of water with the body immersed in it, subtract the value of the volume of water before the body is immersed. You will get the volume of the body.

Find the volume using a rectangular container. If you used a cuboid container, measure the distance between the two marks (the water level before the body is submerged and the water level after the body is submerged), as well as the length and width of the water container. Find the volume of water displaced by multiplying the length and width of the container, as well as the distance between the two marks (that is, you calculate the volume of a small rectangular parallelepiped). You will get the volume of the body.

Do not measure the height of the water container. Measure only the distance between the two marks.
Use

Geometric figures are closed sets of points on a plane or in space that are limited finite number lines. They can be linear (1D), planar (2D) or spatial (3D).

Any body that has a shape is a collection of geometric shapes.

Any figure can be described mathematical formula varying degrees of complexity. Starting from simple mathematical expression up to the sum of series of mathematical expressions.

The main mathematical parameters of geometric shapes are the radii, the lengths of the sides or faces, and the angles between them.

Below are the main geometric shapes most commonly used in applied calculations, formulas and links to calculation programs.

Linear geometric shapes

1. Point

A point is the base object of a measurement. The main and only mathematical characteristic of a point is its coordinate.

2. Line

A line is a thin spatial object having a finite length and representing a chain of points connected to each other. The main mathematical characteristic of a line is its length.

A ray is a thin spatial object that has an infinite length and is a chain of points connected to each other. The main mathematical characteristics of a ray are the coordinate of its beginning and direction.

Flat geometric shapes

1. Circle

A circle is a locus of points on a plane, the distance from which to its center does not exceed a given number, called the radius of this circle. The main mathematical characteristic of a circle is the radius.

2. Square

A square is a quadrilateral in which all angles and all sides are equal. The main mathematical characteristic of a square is the length of its side.

3. Rectangle

A rectangle is a quadrilateral with all angles equal to 90 degrees (right angles). The main mathematical characteristics of a rectangle are the lengths of its sides.

4. Triangle

A triangle is a geometric figure formed by three segments that connect three points (the vertices of a triangle) that do not lie on one straight line. The main mathematical characteristics of a triangle are the lengths of the sides and the height.

5. Trapeze

A trapezoid is a quadrilateral in which two sides are parallel and the other two sides are not parallel. The main mathematical characteristics of a trapezoid are the lengths of the sides and the height.

6. Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel. The main mathematical characteristics of a parallelogram are the lengths of its sides and its height.

A rhombus is a quadrilateral in which all sides, and the angles of its vertices are not equal to 90 degrees. The main mathematical characteristics of a rhombus are its side length and height.

8. Ellipse

An ellipse is a closed curve on a plane, which can be represented as an orthogonal projection of a section of a cylinder circle onto a plane. The main mathematical characteristics of a circle are the length of its semiaxes.

Volumetric geometric shapes

1. Ball

A ball is a geometric body, which is a collection of all points in space located at a given distance from its center. The main mathematical characteristic of a ball is its radius.

A sphere is a shell of a geometric body, which is a collection of all points in space located at a given distance from its center. The main mathematical characteristic of a sphere is its radius.

A cube is a geometric body, which is a regular polyhedron, each face of which is a square. The main mathematical characteristic of a cube is the length of its edge.

4. Parallelepiped

A parallelepiped is a geometric body, which is a polyhedron with six faces and each of them is a rectangle. The main mathematical characteristics of a parallelepiped are the lengths of its edges.

5. Prism

A prism is a polyhedron whose two faces are equal polygons lying in parallel planes, and the remaining faces are parallelograms that have common sides with these polygons. The main mathematical characteristics of a prism are the base area and height.

A cone is a geometric figure obtained by the union of all rays emanating from one vertex of a cone and passing through a flat surface. The main mathematical characteristics of a cone are the radius of the base and the height.

7. Pyramid

A pyramid is a polyhedron whose base is an arbitrary polygon, and the side faces are triangles that have a common vertex. The main mathematical characteristics of the pyramid are the base area and height.

8. Cylinder

A cylinder is a geometric figure limited cylindrical surface and two parallel planes that cross it. The main mathematical characteristics of a cylinder are the radius of the base and the height.

You can quickly perform these simple mathematical operations using our online programs. To do this, enter the initial value in the appropriate field and click the button.

This page contains all the geometric shapes that are most often found in geometry to represent an object or part of it on a plane or in space.

Volume Formula necessary to calculate the parameters and characteristics of a geometric figure.

figure volume- This quantitative characteristic space occupied by a body or substance. In the simplest cases, the volume is measured by the number of unit cubes that fit in the body, that is, cubes with an edge, equal to one length. The volume of a body or the capacity of a vessel is determined by its shape and linear dimensions.

Volumes of geometric shapes.

Figure	Formula	Drawing
Parallelepiped. Volume of a cuboid
Cylinder. The volume of a cylinder is equal to the product of the area of the base and the height. The volume of a cylinder is equal to the product of pi (3.1415) times the square of the radius of the base times the height.
Pyramid. The volume of the pyramid is equal to one third of the base area S (ABCDE) multiplied by the height h (OS).
Correct pyramid- this is a pyramid, at the base of which lies a regular polygon, and the height passes through the center of the inscribed circle to the base.
correct triangular pyramid is a pyramid whose base is equilateral triangle and faces are equal isosceles triangles.
Regular quadrangular pyramid It is a pyramid whose base is a square and whose faces are equal isosceles triangles.
Tetrahedron is a pyramid in which all faces are equilateral triangles.	V = (a 3 √2)/12
Truncated pyramid. The volume of the truncated pyramid is equal to one third of the product of the height h (OS) and the sum of the areas of the upper base S 1 (abcde), the lower base of the truncated pyramid S 2 (ABCDE) and the average proportional between them.	V= 1/3 h (S 1 + √S 1 S 2 + S 2)
Calculating the volume of a cube is easy - you need to multiply the length, width and height. Since the length of the cube is equal to the width and equal to the height, the volume of the cube is s 3 .
Cone- this is a body in Euclidean space, obtained by the union of all rays emanating from one point (the vertex of the cone) and passing through a flat surface.
Frustum obtained by drawing a section parallel to the base of a cone.	V \u003d 1/3 πh (R 2 + Rr + r 2)
The volume of a sphere is one and a half times less than the volume of a cylinder circumscribed around it.
Prism. The volume of a prism is equal to the product of the area of the base of the prism times the height.