A simple task: how to find the perimeter? How to calculate the perimeter of an irregular figure

Surely each of us learned at school such an important component of geometry as perimeter. Finding the perimeter is simply necessary for solving many problems. Our article will tell you how to find the perimeter.

It is worth remembering that the perimeter of any figure is almost always the sum of its sides. Let's look at a few different geometric shapes.

1. A rectangle is a quadrilateral whose parallel sides are equal in pairs. If one side is X and the other is Y, then we get the following formula for finding the perimeter of this figure:

   P = 2(X+Y) = X+Y+X+Y = 2X+2Y.

   An example of solving a problem:

   Let's assume that side X = 5 cm, side Y = 10 cm. So, substituting these values ​​into our formula, we get - P = 2*5 cm + 2* 10cm = 30 cm.

2. A trapezoid is a quadrilateral whose two opposite sides are parallel but not equal to each other. The perimeter of a trapezoid is the sum of all four sides:

   P = X+Y+Z+W, where X, Y, Z, W are the sides of the figure.

   An example of solving a problem:

   Let's assume that side X = 5 cm, side Y = 10 cm, side Z = 8 cm, side W = 20 cm. So, substituting these values ​​into our formula, we get - P = 5 cm + 10 cm + 8 cm + 20 cm = 43 cm.

3. The perimeter of a circle (circumference) can be calculated using the formula:

   P = 2rπ = dπ, where r is the radius of the circle, d is the diameter of the circle.

   An example of solving a problem:

   Let's assume that the radius r of our circle is 5 cm, then the diameter d will be equal to 2 * 5 cm = 10 cm. It is known that π = 3.14. This means that by substituting these values ​​into our formula, we get - P = 2*5 cm*3.14 = 31.4 cm.

4. If you need to find the perimeter of a triangle, then you may encounter a number of problems in doing so, since triangles can have very different shapes. For example, there are acute, obtuse, isosceles, right and equilateral triangles. Although the formula for all types of triangles is:

   P = X+Y+Z, where X, Y, Z are the sides of the figure.

   The problem is that when solving many problems to find the perimeter of this figure, you will not always know the lengths of all sides. For example, instead of information about the length of one of the sides, you can have the degree of an angle or the length of the height of a particular triangle. This will significantly complicate the task, but will not make its solution unrealistic. You can read “” about how to find the perimeter of a triangle, no matter what shape it is.

5. The perimeter of a figure such as a rhombus is found in the same way as the perimeter of a square, because a rhombus is a parallelogram that has equal sides. You can find out how to find the perimeter of a square by reading the article on our website "".

   Now you know how to find the side of the perimeter of the geometric figure you need!

Students learn how to find the perimeter in elementary school. Then this information is constantly used throughout the entire course of mathematics and geometry.

The theory common to all figures

The sides are usually designated by Latin letters. Moreover, they can be designated as segments. Then you will need two letters for each side and written in capitals. Or enter the designation with one letter, which will definitely be small.
Letters are always chosen alphabetically. For a triangle they will be the first three. A hexagon will have 6 of them - from a to f. This is convenient for entering formulas.

Now about how to find the perimeter. It is the sum of the lengths of all sides of the figure. The number of terms depends on its type. The perimeter is designated by the Latin letter R. The units of measurement are the same as those given for the sides.

Formulas for the perimeters of different figures

For a triangle: P=a+b+c. If it is isosceles, then the formula is transformed: P = 2a + b. How to find the perimeter of a triangle if it is equilateral? This will help: P = 3a.

For an arbitrary quadrilateral: P=a+b+c+d. Its special case is the square, the perimeter formula: P = 4a. There is also a rectangle, then the following equality is required: P = 2 (a + b).

What if the length of one or more sides of the triangle is unknown?

Use the cosine theorem if the data includes two sides and the angle between them, which is denoted by the letter A. Then, before finding the perimeter, you will have to calculate the third side. For this, the following formula is useful: c² = a² + b² - 2 av cos(A).

A special case of this theorem is that formulated by Pythagoras for a right triangle. In it, the value of the cosine of the right angle becomes equal to zero, which means that the last term simply disappears.

There are situations when you can find out how to find the perimeter of a triangle by looking at one side. But at the same time, the angles of the figure are also known. Here the theorem of sines comes to the rescue, when the ratios of the lengths of the sides to the sines of the corresponding opposite angles are equal.

In a situation where the perimeter of a figure needs to be determined by its area, other formulas will come in handy. For example, if the radius of the inscribed circle is known, then in the question of how to find the perimeter of a triangle, the following formula will be useful: S = p * r, here p is the semi-perimeter. It must be derived from this formula and multiplied by two.

Sample problems

Condition of the first. Find out the perimeter of a triangle whose sides are 3, 4 and 5 cm.
Solution. You need to use the equality stated above and simply substitute the data into it in the value problem. The calculations are easy and result in a figure of 12 cm.
Answer. The perimeter of the triangle is 12 cm.

Condition two. One side of the triangle is 10 cm. It is known that the second is 2 cm larger than the first, and the third is 1.5 times larger than the first. You need to calculate its perimeter.
Solution. In order to recognize it, you will need to count the two sides. The second is defined as the sum of 10 and 2, the third is equal to the product of 10 and 1.5. Then all that remains is to count the sum of three values: 10, 12 and 15. The result will be 37 cm.
Answer. The perimeter is 37 cm.

Condition three. There is a rectangle and a square. One side of the rectangle is 4 cm, and the other is 3 cm larger. You need to calculate the side of a square if its perimeter is 6 cm less than that of a rectangle.
Solution. The second side of the rectangle is 7. Knowing this, it is easy to calculate its perimeter. The calculation gives 22 cm.
To find out the side of a square, you must first subtract 6 from the perimeter of the rectangle, and then divide the resulting number by 4. The result is the number 4.
Answer. The side of the square is 4 cm.

It is enough to find out the length of all its sides and find their sum. The perimeter is the total length of the boundaries of a flat figure. In other words, it is the sum of the lengths of its sides. The unit of measurement for the perimeter must match the unit of measurement for its sides. The formula for the perimeter of a polygon is P = a + b + c...+ n, where P is the perimeter, but a, b, c and n are the length of each side. Otherwise, it is calculated (or the perimeter of a circle): use the formula p = 2 * π * r, where r is the radius and π is a constant number approximately equal to 3.14. Let's look at a few simple examples that clearly demonstrate how to find the perimeter. As an example, let's take such figures as a square, a parallelogram and a circle.

How to find the perimeter of a square

A square is a regular quadrilateral in which all sides and angles are equal. Since all sides of a square are equal, the sum of the lengths of its sides can be calculated using the formula P = 4 * a, where a is the length of one of the sides. Thus, with a side of 16.5 cm it is equal to P = 4 * 16.5 = 66 cm. You can also calculate the perimeter of an equilateral rhombus.

How to find the perimeter of a rectangle

A rectangle is a quadrilateral whose angles are all 90 degrees. It is known that in a figure such as a rectangle, the lengths of the sides are equal in pairs. If the width and height of a rectangle are the same length, then it is called a square. Typically, the length of a rectangle is the largest side, and the width is the smallest. Thus, to get the perimeter of a rectangle, you need to double the sum of its width and height: P = 2 * (a + b), where a is the height and b is the width. Having a rectangle, one side of which is long and equal to 15 cm, and the other wide with a set value of 5 cm, we get a perimeter equal to P = 2 * (15 + 5) = 40 cm.

How to find the perimeter of a triangle

A triangle is formed by three segments that connect at points (vertices of the triangle) that do not lie on the same line. A triangle is called equilateral if all three of its sides are equal, and isosceles if there are two equal sides. To find out the perimeter, you need to multiply the length of its side by 3: P = 3 * a, where a is one of its sides. If the sides of the triangle are not equal to each other, it is necessary to carry out the addition operation: P = a + b + c. The perimeter of an isosceles triangle with sides 33, 33 and 44, respectively, will be equal to: P = 33 + 33 + 44 = 110 cm.

How to find the perimeter of a parallelogram

A parallelogram is a quadrilateral with pairs of parallel opposite sides. Square, rhombus and rectangle are special cases of the figure. The opposite sides of any parallelogram are equal, so to calculate its perimeter we use the formula P = 2 (a + b). In a parallelogram with sides 16 cm and 17 cm, the sum of the sides, or perimeter, is P = 2 * (16 + 17) = 66 cm.

How to find the circumference of a circle

A circle is a closed straight line, all points of which are located at equal distances from the center. The circumference of a circle and its diameter always have the same ratio. This ratio is expressed as a constant, written using the letter π and equals approximately 3.14159. You can find out the perimeter of a circle by multiplying the radius by 2 and π. It turns out that the length of a circle with a radius of 15 cm will be equal to P = 2 * 3.14159 * 15 = 94.2477