How is volume different from perimeter and area?
The perimeter of a shape represents the distance around it, the area of a shape is the surface or flat space that the shape covers (in 2D/two-dimensional shapes) while the volume of a shape is the space it occupies in real life (in 3D/three-dimensional shapes).
To understand the concepts of perimeter, area and volume, let’s take a closer look at the key differences between each:
Definition of perimeter
The perimeter of a shape is the total length of its figure, also known as the sum of the length of all its sides. Perimeter is measured by adding up the length of its numerous sides. Perimeter is always measured in linear units.
For example, if we are looking to calculate the perimeter of a rectangle, we must start by adding up the length of its four sides. Since the parallel sides of a rectangle are always equal, you only need to find the length of two sides.
Definition of area
The area of a shape is the amount of space that is inside its sides or edges (i.e., the size of the flat, the area occupied by a closed shape). The area is measured in square units and is usually calculated by multiplying the shape’s length by its width. However, depending on different polygons, it can have different formulas.
For example, if we are looking to calculate the area of a square, we need to multiply its length by its width. Since the sides of a square are all equal, we need to multiply the size of one side by itself.
Definition of volume
The volume of a shape is the amount of space that the 3D shape occupies. The volume is measured in cubic units due to the fact that it is calculated by multiplying the shape’s area by its height.
Since the height is always evaluated in regular units of measurement (metres, centimetres…) and the area in square units (square metres, square centimetres…), volume is given in cubic units (cubic metres, cubic centimetres…).
For example, if we are looking to calculate the volume of a cylinder, we have to multiply the area of the base (in this case it’s the area of one of the circles) by the height of the cylinder. The height of a cylinder is the distance between its two circles and the area is either given or must be calculated.
How do you find the perimeter of a geometric shape?
To calculate the perimeter of a geometric shape, you have to add up the length of all its sides. If we have a rectangle with the length a and the width b, the formula for the perimeter P is :
P= a+a+b+b
And since the rectangle has two parallel sides, the formula becomes
P= 2a+2b
P= 2(a+b)
Example 1
The length and width of a rectangle are 8 cm and 3 cm, respectively.
The perimeter of this rectangle is P = 8+8+3+3 = 2 x (8+3) = 22 cm
Now, suppose we have a square. The formula for the perimeter P of this square is :
P = a+a+a+a
P = 4a
Example 2
The sides of a square are 7 cm.
The perimeter of this square is P = 7+7+7+7 = 4 x 7 = 28 cm.
Example 3
To measure trickier shapes, such as the circumference of circle (its perimeter), we need the radius of the circle. This is 5 cm.
The perimeter of this circle is C = 2 πr = 2 x π x 5 = 31.42 cm.
How do you find the area of a geometric shape?
To calculate the area of a geometric shape, you have to multiply its length by its width. Let’s take the same figures to avoid any confusion:
If we have a rectangle with the length a and the width b, the formula for the area A of this rectangle is:
A = base × height or A = a x b
Example 1
The height of the rectangle is 12 cm and the width is 5 cm.
The area of the rectangle is A = 12 x 5 = 60 cm²
PS: Don’t forget to add the square unit (2) when calculating the area.
Now, suppose we want to calculate the area of a square, the formula for the area A is :
A = a x a
A = a²
Example 2
The sides of the square we want to study are 9 cm each.
Therefore, the area of the square is A = 9 x 9 = 92 = 81 cm².
Example 3
The base of an equilateral triangle is 7.5 cm and height is 6.5 cm.
Therefore, the area of the triangle is A = (7.5 × 6.5)÷2 = 24.38 cm²
How do you find the volume of a geometric shape?
Moving on to the volume. To calculate the volume of a geometric shape you have to multiply the shape’s area by its height.
To illustrate this, we are going to stick with the same figures and calculate the volume of a right rectangular prism and a cube. We are also going to calculate the volume of a cylinder to clarify the formula even more:
For a right rectangular solid where the area A is given and the height is h, the formula to calculate the volume V is:
V = A x h
Example 1
The area and height of a right rectangular prism are 38 cm² and 6 cm, respectfully.
The volume of this rectangular solid is V = 38 x 6 = 228 cm³
P.S : Don’t forget to add the cubic unit (3) when calculating the volume.
For a cube, it’s a little bit special. If the area is A and the height h, the formula for the volume of the cube V is :
V = A x h
But, since it’s a cube, the sides are equal to its height so the area is b². The formula for the volume V of the cube can therefore be expressed as follows :
V = b² x b
V = b³
Example 2
The cube has sides of 12 cm each.
The volume of this cube is V = 123 = 1728 cm³
For a cylinder, let’s suppose the area A of the circle was given as well as its height h. The formula of the volume V of the cylinder is :
V = A x h
If the area was not given, we must calculate it first. So the formula becomes :
V = A x h
V = π x r² x h
… with r being the radius of the circle.
Example 3
Our cylinder has a height of 9 cm and a circle radius of 4 cm.
To find the volume of this cylinder, we must find the area first, which in this example is:
A = π x r² = 3.14 x 4² = 50.24 cm²
Now, we can calculate the volume of the cylinder:
V = A x h = 50.24 x 9 = 452.16 cm³
PS: Keep in mind that the numbers stated in all of these examples were in the same units, so if they weren’t, it would be important to convert them.
Need help?
Do you need help finding the perimeter or area of compound shapes? Do you find it difficult to differentiate a 2-dimensional shape from a 3-dimensional shape, or from any of the other above-mentioned shapes? With a variety of exercises and techniques, our math tutors can help you.
FAQ
What is the formula of the perimeter area?
To measure the perimeter of a basic shape, you need to add up the measurement of all the sides. For a quadrilateral, it would be P = s + s + s + s.
What is the formula for the area?
The formula for the area depends on the flat shape. For example, for a rectangle, the formula would be A = base × height, while for a triangle it would be A = (base x height)/2. Don’t forget the answer must be in square units.
What is the volume formula?
The formula for volume depends on the shape. For example, for a cube, the formula would be V = C3, and for a pyramid, V = Ab x H/3.
What is a volume, simple definition?
Volume is all the space occupied by a 3D shape, its content.
Are perimeter and area the same?
The perimeter is the total measurement of all the contours of a 2D shape, while the area is the space occupied by the 2D shape. They have distinct units of measurement.
What are the main differences between perimeter, area and volume?
| Perimeter | Area | Volume | |
| Units of measurement | Regular units of measurement | Square units | Cubic units |
| Types of shapes | 2 dimensional shapes | 2 dimensional shapes | 3 dimensional shapes |
| Examples | If a rectangle:
P = side + side + side + side |
If a rectangle:
A = base x height |
If rectangular prism:
V = length x width x height |
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