Area
The space occupied by a flat shape or the surface of an object is known as its area. The number of unit squares that cover the surface of a closed figure is the area of the figure. Square centimetres, square feet, square inches, and other similar units are used to measure area. The SI unit of area is square meter \((m^2)\), which is a derived unit.
Volume and its Units
The volume of an object is the amount of space it takes up.
In other words, just as height and width are ways to describe size, volume is a measure of an object’s size. The volume of an object is the amount of fluid it can hold if it is hollow (that is, empty). The cubic metres \((m^3)\) the SI unit of volume and is a derived unit. A cubic decimeter\((dm^3)\)is referred to as a litre (L).
So, we see that surface area is a square function whereas volume is a cubic function.
Unit Cubes
A unit cube is a cube with sides that are one unit long each. A 3-dimensional unit cube has a volume of 1 cubic unit and a total surface area of 6 square units.
Cubic units can be defined as units of volume measurement in geometry.
A cubic unit is the volume of a unit cube whose length, width, and height are all one unit.
What is a Rectangular Prism?
A rectangular prism is a three-dimensional shape with one top face, one bottom face, and four lateral sides. All of the prism’s faces are rectangular. A rectangular prism is also known as a cuboid because of its shape. A geometry box, notebooks, diaries, and television remotes are examples of objects that are in the form of a rectangular prism. The shape of a rectangular prism can be seen in the diagram below:
The Properties of a rectangular prism:
The properties of a rectangular prism are listed below to make them easier to recognize by the student:
- A rectangular prism has six faces, eight vertices, and twelve edges.
- The faces of a right rectangular prism are in the shape of a rectangle, whereas the faces of an oblique rectangular prism are parallelograms.
- A rectangular prism has three dimensions: length, width, and height.
- The opposite faces of a rectangular prism are congruent.
Note: Congruent refers to the same size and shape.
How do you calculate the Volume of a Rectangular Prism?
Before using the formula to calculate the volume of a rectangular prism, make sure that all of the dimensions are in the same units. The volume of a rectangular prism is calculated using the steps below:
Step 1: Determine the type of base and its area using an appropriate formula.
Step 2: Determine the height of the prism, which is the measure of the perpendicular from the top vertex of the prism to its base.
Step 3: To calculate the volume of the rectangular prism in cubic units, multiply the base area by the height of the prism.
So, the volume of a prism is equal to the base area multiplied by the height of the prism.
The volume of a rectangular prism = width\(\times\)height\(\times\)length
The volume of a rectangular prism = whl
Where w, h, and l represent the width, height and length of the prism, respectively.
The Formula for the Volume of a Rectangular Prism
The volume of a rectangular prism is used to calculate the number of occupied units. The volume of a rectangular prism is measured in cubic units.
The area of the base multiplied by the height equals the volume of the rectangular prism.
The volume of a rectangular prism is calculated using the formula volume = width\(\times\)height\(\times\)length. Its volume can be written as \(V=\text{width}\times\frac{\text{height}}{2}\times 2(\text{length})\) if the length is doubled and the height is cut in half. After simplifying, we get w\(\times\)h\(\times\)l, which is the standard formula for volume.
What is the Base Area?
The base area refers to the area of the base of a solid figure. It can be used to determine the volume of solid figures. The volume of a prism is calculated using the formula V = B x h, where V is the volume, B is the base area, and h is the height.
For a rectangular prism, the base is rectangular in shape.
Let us take an example where the length of the rectangular base of a prism is 5 cm and its width is 2 cm.
A = l\(\times\)w is the area of a rectangle with length l units and width w units.
So, the base area is 5\(\times\)2, or 10 \(\text{cm}^2\).
Solved Volume of Rectangular Prism Examples
Example 1: Find the volume of the prism shown in the figure.
Solution:
V = B × h Volume of a prism formula
= (l × b) × h Area of a rectangle formula
= (6 × 8 )× 15 Substitute values for l, b, and h
= 720 Simplify
The volume of the prism is 720 cubic yards.
Example 2: Find the missing dimension of the rectangular prism.
Solution:
V = B × h Volume of a prism formula
V = (l × b) × h Area of a rectangle formula
72 = (l × 2) × 3 Substitute the value of the volume and dimensions
72 = 6l Simplify
l = 12
The length of the given rectangular prism is 12 inches.
Example 3: A rectangular aquarium measures 500 mm in length and 100 mm in width. The water level in the aquarium rises by 50 mm when a few fish are put in it. Determine the volume of the fish.
Solution:
The volume of the fish = the volume of the water displaced.
The volume of the fish = (500 × 100 × 50 ) \(\text{mm}^2\)
= (2.5 × \(10^6)~\text{mm}^2\)
Volume is measured in cubic units because it occupies three-dimensional space. One unit of length is used to measure distance (for example, meters). Since it occupies a two-dimensional space, the area is measured in the second unit of length (for example, square metres). The two dimensions are perpendicular. Each of the three dimensions used to calculate the volume is perpendicular to the others. The three dimensions are multiplied to find the volume, for example, cubic metres.
The capacity of a container refers to the amount of liquid it can hold. The most common units of measurement are litres, gallons, and pounds. Volume exists in both solid and hollow objects.
An object’s volume is the amount of space it occupies. The most common units (measurement units) are \(cm^3\) and \(m^3\). The capacity is only available to hollow objects.
The pointy bits or corners where edges meet are called vertices. The lines that surround a shape are called edges. Faces are the flat sides of a shape that you touch while holding it.
An oblique prism is one with bases that are not perpendicular to one another. In other words, an oblique rectangular prism is a rectangular prism with bases that are not aligned one above the other.