Perimeter
In geometry, the perimeter (from the Greek περί- [peri-], 'around', and -μετρος [-meters], 'measure') is the sum of all sides.
A perimeter is a closed path that encompasses, surrounds, or skirts a two-dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Practical applications
The perimeter and the area are fundamental magnitudes in the determination of a polygon or a geometric figure; is used to compute the border of an object, such as a farm fence or piece of land. Area is used when we can get the inside surface of a perimeter that we want to cover with something, such as grass or fertilizer.
Polygons
Regular polygons are needed to determine perimeters, thus not only because they are the simplest shapes, but also because the perimeters of many shapes are computed by approximating them.
The first mathematician known for having used this type of reasoning is Archimedes, which approaches the perimeter of a circle surrounding it with regular polygons. The perimeter of a polygon is equal to the sum of the lengths of its sides. In particular, the perimeter of a rectangle of width a{displaystyle a} and length l{displaystyle l} equals 2a+2l{displaystyle 2a+2l}. An equilateral polygon is a polygon that has all the sides of the same length (for example, a rombo is a 4-sided equilateral polygon).
To find the perimeter of an equilateral polygon, multiply the common length of the sides by the number of sides. A regular polygon can be defined by the number of its sides and by its radius, that is, the constant distance between its center and each of its vertices.
Equations
Perimeter of a polygon
The perimeter of a polygon is calculated by adding the lengths of all its sides. Thus, the formula for triangles is P = a + b + cWhere a{displaystyle scriptstyle a}, b{displaystyle scriptstyle b} and c{displaystyle scriptstyle c} are the lengths of each side. For quadrilateral, the equation is P = a + b + c + d. More generally, for a polygon n{displaystyle scriptstyle n} sides:
P=a1+a2+a3+...+an=␡ ␡ i=1nai{displaystyle P=a_{1}+a_{2}+a_{3}+a_{n}=sum _{i=1}^{n}{a_{i}}}}}}}}
where n{displaystyle scriptstyle n} is the number of sides and ai{displaystyle scriptstyle a_{i}} is the length of the side i{displaystyle scriptstyle i}.
For an equilateral or regular polygon, that is, with all sides equal:
P=na{displaystyle P=na}
where n{displaystyle scriptstyle n} is the number of sides and a{displaystyle scriptstyle a} It's the length of the side.
Circles
The perimeter of a circle is a circumference and its length is:
P=2π π ⋅ ⋅ r=dπ π {displaystyle P=2pi cdot r=dpi }
where:
- P{displaystyle P,} is the length of the perimeter
- π π {displaystyle pi ,} is the constant math pi (π π =3.141592653...{displaystyle pi =3.141592653})
- r{displaystyle r,} is the radius length
- d{displaystyle d,} is the diameter length
To obtain the perimeter of a circle, multiply the diameter by the number π.
Semicircle
A semicircle is delimited by a diameter and half a circumference, so its perimeter is:
- P=2r+r⋅ ⋅ π π =r(2+π π ){displaystyle P=2r+rcdot pi =r(2+pi)}
or
- P=d+(d⋅ ⋅ π π )/2=d(1+π π /2){displaystyle P=d+(dcdot pi)/2=d(1+pi /2)}
where:
- P{displaystyle P,} is the length of the perimeter
- π π {displaystyle pi ,} is the constant math pi (π π =3.14159...{displaystyle pi =3.14159... !)
- r{displaystyle r,} is the radius length
- d{displaystyle d,} is the diameter length
General formulas
| shape | formula | variables |
|---|---|---|
| circle | 2π π r=π π d{displaystyle 2pi r=pi d} | where r{displaystyle r} It's the circle radius and d{displaystyle d} The diameter |
| triangle | a+b+c{displaystyle a+b+c,} | where a{displaystyle a}, b{displaystyle b} and c{displaystyle c} are the lengths of the sides of the triangle. |
| square/rombo | 4l{displaystyle 4l} | where l{displaystyle l} It's the length of the side. |
| Rectangle | 2(l+a){displaystyle 2(l+a)} | where l{displaystyle l} is the length and a{displaystyle a} the width. |
| polygon equilateral | n× × l{displaystyle ntimes l,} | where n{displaystyle n} is the number of sides and l{displaystyle l} is the length of one side. |
| regular polygon | 2newithout (π π n){displaystyle 2nesin left({frac {pi }{n}}}{right)} | where n{displaystyle n} is the number of sides and e{displaystyle e} the distance between the center of the polygon and one of the vertices of it. |
| polygon | a1+a2+a3+ +an=␡ ␡ i=1nai{displaystyle a_{1}+a_{2}+a_{3}+cdots +a_{n}=sum _{i=1}{n}a_{i}}} | where ai{displaystyle a_{i}} is the length of the side i{displaystyle i} (1.o, 2.o, 3.o... No.) side of a polygon of n sides. |
