Interestingly, many years ago, such a branch of mathematics as “geometry” was called “land surveying”. And how to find the perimeter and area has been known for a long time. For example, they say that the very first calculators of these two quantities are the inhabitants of Egypt. Thanks to such knowledge, they could build the structures known today.
The ability to find the area and perimeter can be useful in everyday life. In everyday life, these values are used when it is necessary to paint, plant, or process a garden, stick wallpaper in a room, etc.
Most often you need to know the perimeter of polygons or triangles. To determine this value, it is enough to know the lengths of all sides, and the perimeter is their sum. To find the perimeter, if the area is known, is also possible.
If you need to know the perimeter of a triangle, to calculate it it is worth applying the following formula P = a + b + c, where a, b, c are the sides of the triangle. In this case, all sides of an ordinary triangle on the plane are summed.
The perimeter of a circle is usually called the circumference. To find out this value, it is necessary to use the formula: L = π * D = 2 * π * r, where L is the circumference, r is the radius, D is the diameter, and the number π, as is known, is approximately equal to 3.14.
The formulas for the perimeters of the square and the rhombus are the same, because both sides are equal on one figure and the other. Since the square and the rhombus have equal sides, they (sides) can be denoted by a single letter “a”. It turns out that the perimeter of the square and rhombus is equal to:
- P = a + a + a + a or P = 4a
The perimeter of a rectangle - by what rules is it found?
In order to find the desired value, you need to remember what is called the perimeter - and what features rectangles have.
- The definition of the perimeter sounds like this - this is the sum of the lengths of all sides folded together. The indicator is written with the letter R.
- Moreover, the rectangle is characterized precisely by the fact that those of its sides that are parallel to each other are absolutely equal.
Finding the perimeter of a triangle is a very simple task. It is enough to know only the indicators of the length of two sides, and the remaining two sides will have the same values.
There are two formulas for calculating the perimeter:
- the addition of all sides - in this case, in the abstract rectangle ABCD, the sides AB, BC, CD and AD sequentially fold together and get the result,
- addition of length and width and multiplication of the sum by 2 - here the rule of equality of parallel sides in a rectangle is used.
In addition, you need to remember the special case of a square, when all four sides are equal to each other. Then the length of one side is enough to simply multiply by 4.
The opposite sides of the rectangle and parallelogram are the same, so they can be identified by two different letters “a” and “b”. The formula is as follows:
- P = a + b + a + b = 2a + 2b. Two can be put out of brackets, and we get the following formula: P = 2 (a + b)
The trapezium has different sides, so they are denoted by different letters of the Latin alphabet. In this regard, the formula for the trapezoid perimeter looks like this:
- P = a + b + c + d Here all sides are summed up together.
You can learn more about calculating the perimeter from the article How to find the perimeter.
The area is that part of the figure that is enclosed within its outline.
Rectangle Area - Formula
It is not much harder to calculate the area of a geometric figure. The area is usually denoted by the letter S, and it is measured in square centimeters, millimeters or meters - in contrast to the perimeter, where just meters, millimeters and centimeters are used.
S = a * b, so to find the area you just need to know the length of the rectangle and its width - that is, the indicators of two of the sides. They need to be multiplied among themselves and record the answer in the specified units of length.
The formula for finding the area of a square looks even simpler. Since the sides of the geometric shape are equal to each other, the length and width will be the same. It is necessary to take the indicator of one of the parties and square it. It is written as follows - S = a2.
When recording the solution to the problem of finding the perimeter or area next to the designations P or S, it is customary to put small letter designations of a specific figure. For example, Pabcd, or Sabcd. This allows you to remember for which particular rectangle the area or perimeter is sought.
To calculate the area of a rectangle, it is necessary to multiply the value of one side (length) by the value of the other (width). If the values of length and width are indicated by the letters "a" and "b", then the area is calculated by the formula:
As you already know, the sides of the square are equal, therefore, to calculate the area, you can simply take one side into the square:
The formula for finding the area of a rhombus has a slightly different form: S = a * hawhere ha Is the length of the height of the rhombus that is drawn to the side.
In addition, the area of the rhombus can be found by the formulas:
- S = a 2 * sin α, while a is the side of the figure, and the angle α is the angle between the sides,
- S = 4r 2 / sin α, where r is the radius of the circle inscribed in the rhombus, and the angle α is the angle between the sides.
The circle area is also easily recognizable. To do this, you can use the formula:
- S = πR 2, where R is the radius.
To calculate the area of the trapezoid, you can use this formula:
- S = 1/2 * a * b * h, where a, b are the bases of the trapezoid, h is the height.
To calculate the area of this figure, you must substitute the values in one of the formulas:
- S = a * b * sin α (where a, b are the bases of the parallelogram, α is the angle between the sides),
- S = a * ha (where a is the side of the parallelogram, ha Is the height of the parallelogram, which is lowered to side a),
- S = 1/2 * d * D * sin α (where d and D are the diagonals of the parallelogram, α is the angle between them).