In geometry, the perimeter is simply the total length around a shape. Whether you’re preparing for the SAT, ACT, PSAT, or even the MCAT, understanding how to calculate the perimeter of different shapes is crucial. For example, the perimeter of a circle is calculated using the formula Circumference = 2πr, where r is the radius. For a triangle, you add up the lengths of all three sides with the formula Perimeter = a + b + c. A square’s perimeter is found by multiplying the side length by 4 (Perimeter = 4s), and a rectangle’s perimeter is calculated with Perimeter = 2(l + w), where l is the length and w is the width. For shapes like a rhombus or parallelogram, similar formulas are used, based on the side lengths. Mastering these will help you ace perimeter questions on any of these exams.

Perimeter

The perimeter meaning can be understood from the fact that ‘perimeter’ is a combination of two terms namely; ‘peri’ which implies around, and ‘metron’ which signifies measure. Perimeter is defined as the length of the boundary enclosing a geometrical figure. The perimeter incase of a polygon is the total of all the sides added together. For other figures like circles and ellipse we need to find the length of the boundary of the object. In other words, perimeter is the total length of the border/edge of any closed shape object.

For example, the below image shows a rectangular football pitch. The perimeter here would be the total length surrounding the pitch, i.e, the sum of the lengths of its four sides.

How to Find the Perimeter

Now that we have the perimeter definition, let us understand the different approaches involved in the calculation for the same:

Approach 1: The basic approach to measuring the perimeter of some small shapes/objects like triangles, squares, rectangles, parallelograms, etc is by using a ruler. By using the ruler measure the length of each side of the given shape and sum them together for the perimeter.

Approach 2: The next approach is to use a thread or string. In this method take a thread/string and place it around the boundary of the object and mark the start and end point. Later measure the length of the thread from start to end mark using a scale.

Approach 3: Another approach for the calculation of perimeter will be using the formula for the perimeter of 2D geometric shapes. These formulas are derived from the properties of these shapes.

Perimeter Formula

The perimeter formula for the different geometric shapes varies as per the measure of their dimensions. The approach to finding the perimeter of the different types of polygons is related as we need to sum the length of its sides. In the case of a circle and an ellipse, the perimeter is called the circumference.

The perimeter formula for all shapes are listed below:

Units of Perimeter

Perimeter, as well as the area, are the two important parameters of a 2D shape. Here, we are talking about the perimeter and it is the distance covered around the two-dimensional figure. Thus, its unit is the same as the unit of length. Hence the length of the perimeter is measured in centimeters, kilometers, meters, feet, inches, yards, miles and so on.

Semi Perimeter

Semi perimeter means half of the perimeter of a shape. Thus it is half the length of the boundary of any given two dimensional shape. For any polygon it would be given \(\frac{\text {sum of the sides}}{2}\). For a circle the semi perimeter is half the circumference; \(\pi \times r\). The semi perimeter of a triangle is used to find it area via the Herons formula; \(\text{Area}=A=\sqrt{s(s-a)(s-b)(s-c)}\). Here, “s” is semi-perimeter of the triangle and is equal to; \(\frac{\left(a+b+c\right)}{2}\).

Difference between Area, Circumference and Perimeter

The area, circumference as well as perimeter are the three most common measures of 2D geometry. Hence, let us understand the common difference between them.

Applications of Perimeter

As of now we read about the perimeter meaning, how to find it with the formula and perimeter vs area comparison. Let us understand the uses of the same.

• The perimeter is used to determine the space surrounding a playground, garden, floors and many such objects.
• It is also used in preparing the design for the construction of a house/apartment or a swimming pool.
• In mathematics as well as geometry it helps to obtain the measurement of two-dimensional figures – parallelograms, triangles, rectangles, squares, rhombus, circles and so on.
• It is also used when we are needed to find the amount of net required to fence the lands/plot, the dimension of a chamber’s walls for painting purposes, etc.
• The perimeter along with the area holds applications in mapping and navigation as well.

Solved Examples of Perimeter

Now that you know the difference between circumference and perimeter along with the different uses and formulas. Let us practise some solved examples related to the topic:

Solved Example 1: For a regular octagon tile, the measure of the side is 4.3 cm. Obtain the perimeter for the same.

Solution: The length of the octagon tile is 4.3 cm.

We are having a regular octagon, hence the measure of all the sides will be equal. Moreover, an octagon has eight sides.

Therefore, the perimeter of the regular octagon = 8a (‘a’ is the side of the octagon)

Perimeter = 8 × 4.3 = 34.4 cm

Solved Example 2: Determine the perimeter of a rectangular-shaped study table if the length and breadth of the study table are 32 m and 12 m respectively?

Solution: 

Given;

Length of the rectangular study table= 32 m

The breadth of the rectangular study table=12 m.

Perimeter of a rectangle = 2(length+breadth)

Perimeter of the rectangular study table= 2(32 + 12) = 88 m.

Solved Example 3: What is the effect on the perimeter of a square box, if all its sides are increased by 3 cm?

Solution: 

The perimeter of a square is calculated using the formula; 4 ✕ sides.

Hence, if each side increases by 3 cm, the effect on the perimeter will be:

Perimeter=4 ✕ sides=4 ✕ (x+3)=4x+12

Here x is the side length of the square.

The perimeter will increase by; 12+4x cm.

Solved Example 4: A dairy milk bar is made up of 8 small square pieces(arranged in 2 rows and 4 columns). The side of each square is 2 cm. Calculate the perimeter of the complete square bar.

Solution: Each small square piece has a side length of 2 cm. The eight pieces are arranged in 2 rows and 4 column makes a rectangle.

The length of the rectangular dairy milk(4 square pieces)=4x 2cm=8cm

The breadth of the rectangular dairy milk(2 square pieces)=2x 2cm=4cm

Perimeter of the rectangular dairy milk=2(length+breadth)

=2(8+4)cm

=24 cm

Solved Example 5: If the perimeter of an equilateral triangle is 24 cm. Determine the side length of the triangle?

Solution:

All the sides of an equilateral triangle are of equal length.

Thus, The perimeter of an equilateral triangle is given by the formula = 3 ✕ length of a side

24 = 3 ✕ length of a side

The length of each side = 8 cm

As the length of every side in an equilateral triangle is the same, the length of each side=8 cm.

Solved Example 6: The radius of a circular disk is 42 cm, Obtain the perimeter of the disk.

Solution:

Radius of circular disk = 42 cm

Perimeter of the circle = Circumference of the circle = 2πr

Hence;

Perimeter = 2 × 22/7 × 42

= 2 × 22 × 6

Perimeter of circular disk= 264 cm.

Conclusion

In conclusion, understanding perimeter is essential for solving geometry problems, especially in exams like the SAT, ACT, or MCAT. Whether it's a square, triangle, or circle, knowing the right formula helps you quickly calculate the boundary of any shape. By practicing with real-life examples, you can master these calculations and boost your confidence in tackling geometry questions. So, keep these formulas handy and keep practicing!