Area and Perimeter of Geometric Figures

Area and perimeter are two of the most important measurements of geometric figures. The perimeter represents the sum total of the lengths of the sides of a geometric figure. On the other hand, the area represents the region covered by the figure. The perimeter is a one-dimensional measure and the area is a two-dimensional measure, so the perimeter can be measured in m, cm, etc, while the area can be measured in m², cm², etc.

Here, we will learn about the formulas for the area and perimeter of important geometric figures. Then, we will apply these formulas when solving some problems.

GEOMETRY
height of a scalene triangle

Relevant for

Learning about the area and perimeter of geometric figures with examples.

See formulas

GEOMETRY
height of a scalene triangle

Relevant for

Learning about the area and perimeter of geometric figures with examples.

See formulas

Definition of area

The area of a geometric figure is defined as the region covered by the figure. The area is a two-dimensional measure, so we use square units like m² or cm² to measure it. The area formula depends on the shape of the geometric figure. Depending on the shape of a figure, we will need different dimensions to find its area.

For two geometric figures to have the same area, they must have the same shape and dimensions. For example, we can imagine that we have a rectangle that has a width of length A and a base of length B.

Now, let’s imagine that we have a second rectangle with a width of length C and a base of length D. Therefore, for the area of these two rectangles to be the same, we must have A=C and B=D.


Definition of perimeter

The perimeter of a geometric figure is the total distance around its limits. The perimeter is calculated by adding the lengths of all the sides of the figure. Since the perimeter is a length, we use one-dimensional units like meters or centimeters to measure it.

Due to the nature of the perimeter, it is possible that two figures that have different shapes may have the same perimeter depending on the dimensions of their sides. For example, it is possible to form a circle using a rope and then use the same rope to form a square.

The perimeter formula is different for different geometric figures depending on the number of sides and the shape of the figure.


Formulas for the area and perimeter of geometric figures

There are a great variety of geometric figures, so we need several formulas to calculate their area and perimeter. However, we can become familiar with the formulas of the most common geometric figures, which are the square, the triangle, the rectangle, and the circle.

Formulas for a rectangle

A rectangle is defined as a four-sided 2D figure with right angles. All interior angles in a rectangle measure 90° and their opposite sides are parallel and equal.

rectangle with dimensions

Formula for the Area = a×b

Formula for the Perimeter = 2(a+b)

where a represents the height of the rectangle and b represents its base.

Formulas for a square

A square is defined as a 2D figure that has four sides of equal length. A square is a special type of rectangle since all of its interior angles measure 90°.

diagram of a square with dimensions

Formula for the Area = l²

Formula for the Perimeter = 4l

where l represents the length of one of the sides of the square.

Formulas for a triangle

The triangle is a 2D figure with three sides. There are three types of triangles depending on the lengths of their sides: the equilateral triangle, the isosceles triangle and the scalene triangle. The formulas for calculating the area and perimeter are the same for any type of triangle.

diagram of a triangle with sides and height

Formula for the Area = ½ha

Formula for the Perimeter = a+b+c

where, a, b, c represent the lengths of the sides of the triangle, h represents the height and a represents the base of the triangle.

Formulas for a circle

The circle is a 2D figure that is characterized by having a completely round shape. The radius of the circle is used to calculate its area and perimeter.

diagram of a circle with radius

Formula for the Area = πr²

Formula for the Perimeter = 2πr

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


Table of formulas for the area and perimeter

FigureArea Perimeter
CircleA = πr²P = 2πr
TriangleA = ½ bhP = a+b+c
SquareA = l²P = 4l
RectangleA = abP = 2(a+b)
ParallelogramA = bhP = 2(a+b)
Regular polygonA = ½ nlaP = nl

Circle: r represents the radius and π is the mathematical constant with a value of 3.1415 …

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

Square: l represents the length of one of the sides.

Rectangle: a defines the length of the width and b defines the length of the base.

Parallelogram: b represents the length of the base and h represents the length of the height.

Regular polygon: n defines the number of sides of the regular polygon, l represents the length of one of the sides and a represents the length of the apothem. Recall that the apothem is the perpendicular distance from one of the sides of the polygon to the center.


Area and perimeter – Examples with answers

The formulas for the area and perimeter of different geometric figures are applied to solve the following examples. Each example has its respective solution, where you can see the process used.

EXAMPLE 1

What is the area of a triangle that has a base of 9 m and a height of 8 m?

We know that the area of any triangle can be calculated by multiplying its base and height and dividing by two. Therefore, we have:

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

$latex A=\frac{1}{2}(9)(8)$

$latex A=36$

Therefore, the area of the triangle is 36 m².

EXAMPLE 2

What is the area and perimeter of a square that has sides of length 11 m?

We can find the area by squaring the length of one of the sides. The perimeter is found by multiplying the length of one of its sides by four. Therefore, we have:

$latex A={{l}^2}$

$latex A={{11}^2}$

$latex A=121$

$latex p=4l$

$latex p=4(11)$

$latex p=44$

Therefore, the area of the square is 121 m² and the perimeter is 44 m.

EXAMPLE 3

A rectangle has sides of length 12 m and 15 m. What is its perimeter?

We find the perimeter of the rectangle by adding the lengths of the sides and multiplying by two:

$latex p=2(a+b)$

$latex p=2(12+15)$

$latex p=2(27)$

$latex p=54$

Therefore, the perimeter of the rectangle is 54 m.

EXAMPLE 4

What is the length of the sides of a square that has a perimeter of 60 m?

We have the measure of the perimeter and we want to find the length of one of the sides of the square. We simply use the perimeter formula and solve for one of the sides:

$latex p=4l$

$latex 60=4l$

$latex l=15$

Therefore, the length of one side of the square is 15 m.

EXAMPLE 5

What is the area and perimeter of a circle that has a radius of 8 m?

Both the area and the perimeter of the circle are calculated using the length of the radius:

$latex A=\pi{{r}^2}$

$latex A=\pi{{(8)}^2}$

$latex A=\pi(64)$

$latex A=201.1$

$latex p=2\pi r$

$latex p=2\pi (8)$

$latex p=50.3$

Therefore, the area of the circle is 201.1 m² and the perimeter is 50.3 m.


Area and perimeter – Practice problems

Solve the following problems using what you have learned about the area and perimeter of geometric figures. If you need help with this, you can look at the solved examples above.

What is the perimeter of a square that has sides of length 18m?

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If a circle that has a radius of 7m, what is its area?

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What is the area of a triangle that has a height of 12m and a base of 11m?

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If a rectangle has lengths 13m and 17m, what is its perimeter?

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See also

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