The perimeter of a two-dimensional figure is the length of the boundary of the figure. If the figure is a polygon such as a triangle, square, rectangle, pentagon, etc., then the perimeter is the sum of the edge lengths of the polygon. For example, a square with side length 3 has perimeter
irregular polygon to the right with perimeter Some shapes do not have a finite number of sides, so calculating the perimeter can be less straighforward. For example, the perimeter (or circumference) of a circle is where is the radius of the circle. Fascinatingly, some shapes (such as certain fractals) have infinite perimeter, despite having a finite area.Any closed figure has a perimeter, such as the
What is the perimeter of a square with side length 6?
Since a square has four sides of equal length and the perimeter of a square is the sum of its side lengths, the perimeter of the square is
You can also do on the count that it is a square.
What is the perimeter of a regular pentagon whose side lengths are all 6?
Since a regular pentagon has five sides and all its side lengths are , the perimeter of the pentagon is
What is the perimeter of a rectangle with width and height
Since a rectangle has two sides with equal width and two sides with equal height, and the perimeter of a rectangle is the sum of its side lengths, the perimeter of the rectangle is
Find the perimeter of a triangle with and
First, use the Pythagorean theorem to find the length of
Then add to get the perimeter, which is
Suppose an ant walks along the boundary of a triangle with side lengths and . If the ant starts at one vertex of the triangle and makes complete trips around the triangle boundary to return to its starting point, what is the total distance traveled by the ant?
Since the perimeter is the sum of edge lengths, the perimeter of the triangle is . Since the ant makes complete trips around the triangle boundary, the total distance traveled by the ant is
Find the perimeter of the given figure.
The perimeter of a circle with radius is . The perimeter of an arc of a circle subtending angle (in degrees) at the center is .
What is the perimeter (circumference) of a circle of diameter
Find the perimeter of a semicircle of radius cm.
In the figure to the right, a circle is circumscribed around a regular hexagon, and the same circle is inscribed within another regular hexagon.
Let be the perimeter of the larger hexagon, let be the perimeter of the smaller hexagon, and let be the circumference of the circle.
The circumference of the circle can be approximated by finding the mean of the two perimeters:
If is approximated using the approximation for circumference above, then , where are positive integers, and are coprime, and is square-free.
We can calculate the perimeter of an enclosed figure if its graph on a Cartesian plane is a function in polar coordinates. Specifically, suppose the figure has the equation such that is a continuous, nonnegative function for and . Then its perimeter is given by the following formula:
Suppose we want to calculate the circumference of a circle with radius . Then is a constant, and its derivative is . So the formula gives us
Therefore the circumference is
Unfortunately, for other shapes, this formula usually leads to complicated integrals which often do not have a simple closed form.
In this section, we will learn about applications of the concept of perimeter in real life. They are mostly of fencing-type problems. Here is an example:
Mr. Antony has a rectangular garden of area and length To save his garden from domestic animals grazing them, he wants to fence his whole garden. Find much Mr. Antony has to spend for fencing his field at the rate of $3 per meter?
Given, the area of the rectangle and its length we know that
Now, let's find perimeter:
Since the cost of fencing per meter is $3, the total cost of fencing the whole feild is dollars.