Question

How is the base of a shape related to its volume?

Answer 1

Hint: Here, we will find the volume of the solid in the shape of a Prism and the volume of the solid in the shape of a Pyramid. We will find the relation between the base of a shape and the volume even if the base and height changes, the relation remains the same by using an example in two dimensional shape and three dimensional shape. Thus, the relation between the base of a shape and its volume is the required answer.

Formula Used:
Volume of a cube \[ = {\text{length}} \times {\text{breadth}} \times {\text{height}}\]

Complete Step by Step Solution:
We know that if the solid is in the shape of a Prism whose top and bottom of the solid is of the same size and same shape, then the volume of the solid is given by the product of its base area and the height.
Volume of a Solid in the shape of a Prism, \[V = \] Base area \[ \times \] Height cubic units
We know that if the solid is in the shape of a Pyramid whose top is a single point and has sloping slides, then the volume of the solid is given by the one third of the product of its base area and the height.
Volume of a Solid in the shape of a Pyramid, \[V = \dfrac{1}{3} \times \] Base area \[ \times \] Height cubic units
For example, consider the cube whose side is 1 cm. We will divide the cube into six square based pyramids with a common apex in the center of the cube.
 

Now, we will find the base area of a Pyramid
Base Area of a Pyramid \[ = l \times b{\text{c}}{{\text{m}}^2}\]
\[ \Rightarrow \]Base Area of a Pyramid \[ = 1 \times 1 = 1{\text{c}}{{\text{m}}^2}\]
Thus, the height of the Pyramid \[h = \dfrac{1}{2}{\text{cm}}\]
We know that Volume of a Solid in the shape of a Pyramid, \[V = \dfrac{1}{3} \times \] Base area \[ \times \] Height cubic units
Volume of a square based Pyramid, \[V = \dfrac{1}{3} \times 1 \times \dfrac{1}{2}\]
\[ \Rightarrow \] Volume of a square based Pyramid, \[V = \dfrac{1}{6}\]
We know that Volume of a cube \[ = {\text{length}} \times {\text{breadth}} \times {\text{height}}\].
Now, we can rewrite the volume of a square based pyramid as \[\dfrac{1}{6}\] of the Volume of the cube.
\[ \Rightarrow \] Volume of a square based Pyramid, \[V = \dfrac{1}{6} \times \left( {1 \times 1 \times 1} \right)\]
Thus, A square based pyramid has Volume of \[\dfrac{1}{3}\] of the product of its base area and the height.
The formula remains true even if we stretch or compress the pyramid, uniformly in any one direction, there will be no change in the base and the height so that the volume also remains constant.
Now, we are given any two dimensional shape, then we can approximate it into arbitrarily closed by using the grid of squares, then make square based pyramids such that the squares have a common apex. The total volume is the sum of the volumes of Pyramids, which will be the same for \[\dfrac{1}{3}\] of the height and the sum of the base areas of the pyramids. Since the squares approximate the area of the original shape, the volume of the pyramid which approximates the volume of the pyramid based on the original shape.

Therefore, the volume of a shape is directly proportional to its base.

Note:
We should remember that a pyramid is a solid shape where the sides of the triangle meet at the top which is called the apex and the base is any polygon which is a flat surface. A prism is a solid shape with identical ends, flat faces and the same cross section along its length. A two-dimensional figure has two dimensions length and breadth whereas a three-dimensional figure with three dimensions length, breadth and height. We know that Volume is defined as the quantity of substance that can be contained in an enclosed curve.