How to find the height of a cuboid if given the volume and base area?

Mensuration entails the processes of measurements and all calculations pertaining to various geometrical shapes, occurring in mathematical theory as well as our daily lives. The study of all geometrical shapes falls under the scope of mensuration. Geometric shapes such as triangles, rectangles, quadrilaterals, circles, etc. Here, a cuboid is discussed below,

Cuboid

It is a three-dimensional, box-like shape that can be represented only in the three-dimensional plane. It has six faces, each of which is rectangular in shape. Thus a cuboid is formed when six rectangles are stacked up together, forming all three proportions of a cuboid, i.e., length, breadth, and height. 

Base Area of a Cuboid

A cuboid is composed of 6 faces, all of them being rectangles. The area of a rectangle is the product of its length and breadth. Following this logic, the area of each face of the cuboid can be calculated. Thus, the area of the base of a cuboid would be given by the product of the length and breadth of the rectangle that forms such a base.

Hence, base area of a cuboid = (l × b) sq. units.

Volume of a Cuboid

The volume of any solid represents its capacity. It is the quantification of the space inside a given shape and can only be calculated for three-dimensional shapes.

Volume of cuboid = (l × b × h) cu. units.

How to find the height of a cuboid if given the volume and base area?

Solution:

Let the length, height and breadth of the cuboid be l, b and h units respectively.

Volume of the cuboid = (l × b × h) cu. units and base area of the cuboid = (l × b) sq. units.

Clearly, by taking the ratio of both these parameters, the value of height (h) can be easily found out. Since the common term, i.e., the product of length and breadth would be eliminated. 

Hence, height of a cuboid given its volume and base area = Volume/Base Area.

Example: Say the volume of a cuboid is 300 cu. cm and base area is 150 sq. cm, then its height:

= Volume/ Base area 

= 300/ 150 cm

= 2 cm

Sample Problems

Question 1. Find the height of a cuboid whose volume is 100 cu. cm and the base area is 10 sq. cm.

Solution:

Height of a cuboid given its volume and base area = Volume/ Base Area.

⇒ h = 100/ 10

= 10 cm

Question 2. Find the height of a cuboid whose volume is 690 cu. cm and the base area is 3 sq. cm.

Solution:

Height of a cuboid given its volume and base area = Volume/ Base Area.

⇒ h = 690/3

= 230 cm

Question 3. Find the height of a cuboid whose volume is 420 cu. cm and the base area is 20 sq. cm.

Solution:

Height of a cuboid given its volume and base area = Volume/Base Area.

⇒ h = 420/20

= 21 cm

Question 4. Find the height of a cuboid whose volume is 500 cu. cm and the base area is 40 sq. cm.

Solution:

Height of a cuboid given its volume and base area = Volume/ Base Area.

⇒ h = 500/ 40

= 12.5 cm

Question 5. Find the height of a cuboid whose volume is 69 cu. cm and the base area is 21 sq. cm.

Solution:

Height of a cuboid given its volume and base area = Volume/ Base Area.

⇒ h = 69/ 21

= 3 cm