Perimeter and area are two of the most important properties of two-dimensional figures. The perimeter defines the distance around the limits of the figure, while the area indicates the region occupied by the figure. These properties are used to describe the figure and perform different calculations. The perimeter and area are applicable to any closed figure regardless of whether it is regular or irregular.

Here, we will learn about the formulas for the perimeter and the area of some more common two-dimensional figures. Then, we will use these formulas to solve some problems.

## What is the area?

The area is defined as the region occupied by a figure in two-dimensional space, that is, the area is the space covered by the figure. The area of any figure depends on its dimensions and its properties. This means that the formulas used to calculate the area vary depending on the shape of the figure.

For example, the formula for calculating the area of a triangle is different from the formula for calculating the area of a square.

If two objects have the same shape, their areas will not necessarily be the same. For the area of two objects to be the same, both their shapes and their dimensions must be the same.

For example, suppose we have a rectangle with width A and length B and we have a second rectangle with width C and length D. For the area of the rectangles to be the same, we must have A=C and B=D.

## What is the perimeter?

The perimeter of a figure is defined as the total distance around the figure. The perimeter can be considered as the total length of a figure if it is expanded in a linear way. The perimeter of figures that have different shapes can be the same depending on the dimensions of the figures.

For example, if we build a circle with copper wire, the same copper wire can be used to build a square, which will have sides of equal length.

The perimeter formula will depend on the shape of the figure and the number of sides the figure has.

## Formulas for the perimeter and area of geometric figures

There are a large number of geometric figures, which have different formulas to calculate the perimeter and area. However, the most common shapes are the square, triangle, rectangle, and circle.

### Perimeter and area of a rectangle

A rectangle is a figure in which all its internal angles measure 90° and in which its opposite sides are equal.

Perimeter of a rectangle = **2(a+b)**

Area of a rectangle = **a×b**

where *a* is the width of the rectangle and *b* is its base.

### Perimeter and area of a square

A square is a figure that has four equal sides and whose four interior angles measure 90°. A square is a special type of rectangle.

Perimeter of a square = **4 l**

Area of a square = *l*²

where *l* is the length of one of the sides of the square.

### Perimeter and area of a triangle

A triangle is a figure with three sides. Depending on the characteristics of its sides, the triangle can be equilateral, isosceles, or scalene. However, the formulas for calculating the perimeter and area of the different types of triangles are the same.

Perimeter of a triangle = **a+b+c**

Area of a triangle = **½ha**

where, *a, b, c* are the lengths of the three sides of the triangle, *h* is the height and *a* represents the base of the triangle.

### Perimeter and area of a circle

The circle is a figure that has a completely round shape. Both the perimeter and the area are calculated using the radius, which is the length from the center of the circle to a point on its circumference.

Area of a circle = **πr²**

Perimeter of a circle = **2πr**

where *r* is the radius of the circle and π is a mathematical constant that has a value of 3.1415…

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## Table of formulas for the perimeter and area

Figure | Area | Perimeter |

Circle | A = πr² | P = 2πr |

Triangle | A = ½ bh | P = a+b+c |

Square | A = l² | P = 4l |

Rectangle | A = ab | P = 2(a+b) |

Parallelogram | A = bh | P = 2(a+b) |

Regular polygon | A = ½ nla | P = nl |

Circle: *r* is the radius and π is the constant equal to 3.1415…

Triangle: *b* is the base, *h* is the height and *a, b, c* are the lengths of the sides.

Square: *l* is the length of one side.

Rectangle: *a* is the length of the width and *b* is the length of the base.

Parallelogram: *b* is the length of the base and *h* is the length of the height.

Regular polygon: *n* is the number of sides of the regular polygon, *l* is the length of one of the sides and *a* is the length of the apothem. The apothem is equal to the distance from the center to one of the sides.

## Perimeter and area – Examples with answers

The following examples are solved using the formulas for the perimeter and the area of various geometric figures. Try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

Determine the perimeter of a square that has sides of length 12 m.

##### Solution

A square has four equal sides and the perimeter is equal to the sum of the lengths of all the sides, so we have:

$latex p=4l$

$latex p=4(12)$

$latex p=48$

Therefore, the perimeter of the square is 48 m.

**EXAMPLE 2**

If we have a triangle with a base of 7 m and a height of 10 m, what is its area?

##### Solution

The area of any square is equal to one-half the product of its height and its base. Therefore, we have:

$latex A=\frac{1}{2}bh$

$latex A=\frac{1}{2}(7)(10)$

$latex A=35$

Therefore, the area of the triangle is 35 m².

**EXAMPLE 3**

If the radius of a circle is equal to 7 m, what is the measure of its perimeter?

##### Solution

We use the formula for the perimeter of a circle to find the result:

$latex p=2\pi r$

$latex p=2\pi (7)$

$latex p=44$

Therefore, the perimeter of the circle is 44 m.

**EXAMPLE 4**

A square has a perimeter of 44 m. What is the length of its sides?

##### Solution

In this case, we start from the perimeter and we want to find the length of one of its sides. We use the same formula, but in this case, we have that the perimeter is 44 m, so we solve for the length:

$latex p=4l$

$latex 44=4l$

$latex l=11$

Therefore, the length of each side of the square is 11 m.

**EXAMPLE 5**

If a rectangle has sides of length 12 m and 13 m, what is its area?

##### Solution

To find the area of a rectangle, we simply have to multiply the length of its sides:

$latex A=ab$

$latex A=(12)(13)$

$latex A=156$

Therefore, the area of the rectangle is 156 m².

## Perimeter and area – Practice problems

Use the formulas for the perimeter and area of various figures to solve the following problems. Select your answer obtained and check it to verify that it is correct.

## See also

Interested in learning more about geometric figures? Take a look at these pages:

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