Volumes and contents

First, very briefly on the theory, in case someone doesn't know. The circumference of a solid is the sum of the lengths of all its sides, expressed in metres and derived units. The perimeter is usually denoted by the letter O. The volume of a solid is the size of the area that makes up the solid, it is calculated in square metres, mathematically a square metre is expressed using a two in the superscript: ^m2. The volume is the space that the solid makes up, in simple terms it expresses how much water you can pour in. Volume is calculated in cubic metres and derived units, and spatial measures are written using the three in superscript: ^m3. Volume is commonly written using the letter V. Content and volume can also be expressed in other (and probably more commonly used) units such as ar or hectare for content and litre for volume. For more information on units, see units.cz. Content or volume can generally be calculated by integrals.

Square and rectangle

Both solids are two-dimensional, so only the perimeter and content can be found here. For a square, this is all easiest because a square by definition has all sides the same length. Thus, to calculate the perimeter of a square, we just take one side of the square and multiply it by four (the number of sides): O=4a. The perimeter of a rectangle is only slightly more complicated. A rectangle always has two and two sides of equal length, so there are two ways to go: either simply add all the sides, or take the lengths of two different sides, multiply them by two and add them up: O=2a+2b.

The area of a square is no more complicated than its perimeter. You take one side and multiply by the other side. Since the square has all sides the same length, the formula is: S = a-a = ^a2. A rectangle works exactly the same as a square, except that a rectangle doesn't have all sides the same length, so you have to multiply two different perpendicular sides: S = a-b.

Rectangle

A parallelogram is a figure that is similar to a rectangle but has two opposite sides that are skewed, see the figure below this paragraph. The perimeter of a parallelogram is simple and basically the same as that of a rectangle: O=2a+2b, but the content is a bit more interesting. To calculate the area of a parallelogram, we have to make it a rectangle, otherwise we can't. And the following figure shows us how to make a rectangle out of it:

First, we cut off the excess part of the parallelogram (the coloured part) and add it to the other side of the parallelogram to make a rectangle:

Only now we can easily calculate the area of the parallelogram, the formula will be the same as for the rectangle. However, the graphical conversion of the parallelogram is not convenient, a simple formula is used, based on just that conversion. The side a will be identical in the case of both the rectangle and the parallelogram, only the other side differs. So, for the parallelogram, instead of side b, we will calculate the height of the parallelogram to be equal to side b in the modified rectangle. This is clearly shown in the following figure (we will multiply the sides highlighted in red):

The formula looks like this: _S=va-a.

Odd-lot

The perimeter is clear, let's add all the sides together. The area of the trapezoid is a bigger nut to crack. Theoretically, we could do the same as for the parallelogram, but we don't know whether to take the longer(AB) or the shorter(CD) side.

Therefore, we use a trick - we calculate the average length of the two parallel sides and count with it. So we add the length of the sides AB and CD, divide by two and then calculate the same as for the parallelogram - multiply by the height and we have the content. S=(a+c)/2 -_va.

Triangle

Atriangle is a two-dimensional solid and as such we can calculate its perimeter and content, it cannot and does not have a volume. We calculate the perimeter of a triangle by adding all its sides. Thus, the general formula would look like this: O=a+b+c. Of course, we can find special cases, for example, it is clear that an equilateral triangle has all sides of equal length and so we just need to know the length of one side and multiply that by three. Then the formula would look like this: O'=3a.

We will need one small adjustment for the area of the triangle. In order to calculate the area of a triangle, we need to make it a parallelogram (similar to how we made a rectangle out of a parallelogram). Again, the following little illustration shows us how to make a parallelogram out of a triangle:

From this picture, it should be clear how to calculate the area of a triangle - you'll do the same as if you were calculating the area of a parallelogram, but divide the result by two: S=(_va-a)/2

Circle

A circle is a special shape in that you can't calculate its circumference or content. At least not with an absolutely precise value. Whatever we want to calculate with a circle, we can perhaps never do without the constant Pi - π. The exact value of π is not known, it is an irrational number, but its approximate value, which is usually used (unless you have its value stored in a calculator), is 3.14.

Now let's move on to the formulas. Early thinkers have figured out how to calculate the circumference of a circle, or circle (a circle is just that line, an arc; a circle has no interior and therefore no content, whereas a circle has a content because the interior of the circle is included). It was found that the ratio of the diameter of a circle (which can be easily measured) to the circumference of a circle is always the same, and over time it was even calculated how many times the circumference of a circle is greater than its diameter. And world wonder - it is precisely Pi times. So we calculate the circumference of the circle as O=π-d=2-π-r (the latter, more complicated formula is more often given, because we usually know the radius rather than the diameter), where d is the diameter and r is the radius of the circle.

The volume of the circle is then ^S=π-r2, I don't know what more to say :-).

Cube and cuboid

These are the first spatial solids, so for them we will determine surface area and volume. Since a cube has six faces and all faces are squares, the procedure is clear: you calculate the area of one face and then multiply it by six. ^S=6a2. For a cube, this is already more tedious because it has several different walls. Basically, you just calculate the area of the three different walls, add and multiply by two and you have the surface area of the block: S=(a-b + b-c + a-c)-2.

The volume of a cube is calculated in exactly the same way as the volume of a square, with the slight difference that we must not forget that we are in space. In short, just take the length of the side and multiply it to the third: ^V=a3. For a cube it is similar, but we have to multiply the three sides separately because they are of different lengths: V=a-b-c.

Spheres

Thesurface area of the sphere (=sphere surface) is calculated as ^S=4-π-r2 and the volume of the sphere is ^V=4/3-π-r3. If you want to read a rather complicated derivation of the calculation of the volume of a sphere, have a look at Cavalieri's principle. The derivation of the surface area of a sphere is then, for example, on the forum here.

Cylinders

We calculatethe surface area of a cylinder as the sum of the area of two bases with wall content. The base is a normal circle, so the area of one base will be equal to the area of the circle, so calculate the area of one base as follows:

$$S_p=\pi\cdot r^2$$

The wall of the cylinder is nothing but a "rolled up" rectangle, where the length of one side is equal to the height of the cylinder (denoted by v) and the other is equal to the circumference of the circle at the base. So the area of the wall is equal:

$$S_s=v\cdot2\pi r$$

The surface area of the whole cylinder is thus equal:

$$S=2\cdot S_p+S_s=2\pi r^2+2v\pi r=2\pi r(r+v)$$

The volume of the cylinder is simply the height multiplied by the area of the base. We have already calculated the area of the base in the previous step. Result:

$$V=v\cdot\pi r^2$$

Needle and cone

Thesurface area of a cone is generally calculated as the sum of the area of the base and the area of all the walls. There is not much of a general formula for this. The volume of the cone has something in common with calculating the area of a triangle - you take the area of the base and multiply it by the height of the cone. You then divide the result by three and you have the volume of the pyramid. V = (_Sp - v)/3, where _Sp is the volume of the base.

Thesurface area of the rotating cone is equal to the sum of the base content and the shell content. The area of the base is clear, it is an ordinary circle, so ^S=π-r2. The area of the shell is a bit more complicated, you have to imagine that you "unfold" the shell on the table and this gives you a kind of circular section whose content is equal to: S=π-r-s, where s is the radius of the shell (the distance of the top of the cone from the edge of the base, basically something like the edge of a needle). The volume of the cone is V=(π - ^r2 - h)/3, where h is the height of the cone.

Example for the volumes

Calculate the area of the following figure (assume the measures on the axis are in metres):

At first glance, we find that this solid is a kind of general polygon to which we cannot apply any formula as a whole. We must therefore divide it into smaller parts, which we can already calculate. The most convenient way is to make the bottom part a triangle and the top part a trapezoid. The division is shown in the following figure:

Now calculate the area of triangle ABE using the formula S = (a -_va)/2. The side a in this case will be side AE and the height to this side will be side EB. We see that |AE|=3 m and |EB|=1 m. By substituting into the formula we get: S=(3 - 1)/2=1.5^m2.

In the second step, we calculate the trapezoid's area using the formula S=(a+c)/2 -_va. In our case, side a will be side AE and side c will be side DC. The height is again easily read from the figure, it is the length of the line segment EC. By substituting into the formula we get: S=(3+2)/2 - 2=5^m2.

We just add the previous two partial results and we have the resulting volume of the solid. S = 1.5 m + 5 m = 6.5^m2.

Calculate the area of the following figure (assume that the measures on the axis are in metres):

Again, it is probably obvious at a glance that one pattern will not be enough and that we will have to modify the pattern in some way. We will calculate it in the following way: calculate the area of the "square", add to it three quarters of the area of the circle and subtract the small triangle, bottom left:

We calculate the area of the square simply: _S1 = 2 - 2 = 4^m2. In the second step we calculate the area of the circle. Again, we just add it to the formula and calculate _S2 = π -¹² - 3/4 = 3π/4. All that remains is to calculate the area of the triangle DEI and subtract it from the area of the square. The side EI and its height DI have the same length of 1 m, so the area of the triangle will be _S3 = ½.

In the last step, add and subtract the partial results: S =_S1 +_S2 - _S3 = 3.5^m2 + 3π/4^m2.