How to find the height of a cuboid given the surface area?

Mensuration is a branch of mathematics deals with geometric figure calculation and its different parameters such as length, area, volume, surface area, etc. It deals with the region, size, and density of different dimensions i.e. 2D and 3D. A 2D figure is a shape drawn on a plane and containing two axes (x and y) by three or more straight lines or closed boundaries or segments. Such forms do not have either width or height. whereas a 3 Dimensional shape is a structure bounded by a variety of surfaces or planes. Unlike 2D shapes, these shapes have height or depth and contain all three axes (x, y, and z); they have 3-dimensional length, breadth, and height and therefore these figures are called 3D figures.

Three various 2D shapes form the 3D figures. They contain volume (V), Surface area either Curved surface area (CSA) or lateral surface area (LSA) and total surface area (TSA), etc. Now, let’s discuss some basic terms,

Area (A)

The area of any closed figure is defined as the surface occupied by a defined region. It is generally denoted by the letter A and the unit is m² or cm². In real life, applications of area are seen a lot, for instance, the measurement of lands is done by measuring the area of the land.

Perimeter (P)

The perimeter of any figure is defined as the total length of the boundary of a figure or the sum of all sides is called its perimeter. Perimeter is determined of only 2-dimensional shapes. It is generally represented by P and the unit is m or cm. The perimeter can also be referred to as the periphery of any object and measuring the length of the periphery is referred to as a measurement of the perimeter. For instance, in order to apply a fence around a park, it is important to know the perimeter of the park.

Volume (V)

The volume of 3-D shape is defined as the space that a substance or object occupies, for example, the air contained in the room. Volume is generally denoted by V and the unit of volume is m³ or cm³. An example of volume is obvious, the LPG filled in the cylinders are filled after understanding the quantity that that cylinder withholds, that is, its volume.

Curved Surface Area (CSA)

The curved surface area is the area of the only curved surface, not taking any of the base and top for example sphere or a circle that has a curved surface area. The general representation of curved surface area is CSA and the unit is as of area i.e. m² or cm². It is important to note that only curved surfaces will have CSA, for example, cylinder, sphere, hemisphere, cone, etc.

Lateral Surface Area (LSA)

The lateral surface area is an area of a given figure’s lateral surfaces or sides is called the Lateral Surface Area. Lateral surfaces are the layers covering the outer area. The general representation of the lateral surface area is LSA. The general representation of curved surface area is LSA and the unit is as of area i.e. m² or cm²

Total Surface Area (TSA)

The total surface area is defined as the surface area of all surfaces or the sum of CSA and LSA. For example, it’s the Total Surface Area in a cube by adding the area of all 6 surfaces. The general representation of the lateral surface area is TSA. The general representation of curved surface area is TSA and the unit is as of area i.e. m² or cm²

The general formula for basic shapes

Formulae are generated for the different types of surface areas and perimeters for different shapes. The shapes can be either 2-D shapes or can be 3-D shapes. Let’s take a look at these formulae one by one. They are to be remembered as they are frequently used in calculations in mathematics.

Some 2-D shapes

• Square

1. Area of square (A) =  side × side sq. units
2. The perimeter of Square (P) = 4 × side units

• Rectangle

1. Area of rectangle (A) =  length × breadth sq. units
2. Perimeter of rectangle (P) = 2 × (l + b) units

• Triangle

Below given is the table for different types of triangles and their respective formulae for area and perimeter. The different types of triangles are scalene triangles, equilateral triangles, and isosceles triangles.

Types of triangle	Area (m2)	Perimeter (m)
Scalene triangle	(base × height)/2	side a + side b + side c
Isosceles triangle	(base × height)/2	3 × side
Equilateral triangle	√3/4 × side2	(2 × side) + base

• Circle

1. Area of circle (A) =  π × radius² sq. units
2. Perimeter of circle (P) = 2 π × radius units.

Some of the 3-D shapes

• Cube

1. The volume of cube V = side³ cubic units.
2. Lateral Surface Area of cube LSA =  4 × side² sq. units.
3. Total Surface Area = 6 × side² sq. units.

• Sphere

1. The volume of sphere = 4/3 π × radius³ cubic units.
2. Surface Area of sphere = 4 π × radius² sq. units.

• Hemisphere

1. The volume of hemisphere = 2/3 π × radius³ cubic units.
2. Total Surface Area of hemisphere = 3 π × radius² sq. units.

• Cylinder

1. Volume of cylinder = (π × radius² × height) cubic units.
2. Curved Surface Area of cylinder = (2πRh) sq. units.
3. Total Surface Area of cylinder = (2πRh + 2πR²) sq. units 

Where, R = radius

• Cuboid

1. The volume of the cuboid (V) =length × width × height cubic units.
2. Lateral Surface Area of cuboid (LSA) = 2 × height (length + width) sq. units.
3. Total Surface Area = 2(length × width + length × height + height × width) sq. units.

Cuboid

How to find the height of a cuboid given the surface area?

Solution:

Lets take both the Surface Area i.e. Lateral and total surface area.

Case 1: Lateral Surface Area is given:

Lateral Surface area of Cuboid = 2h (l + b)

Lets assume the LSA to be A.

=> A = 2h (l + b)

=> 2h = A/ (l + b)

=> h = A / 2 (l + b)

So, this formula can be used to calculate height of cuboid 

Case 2: Total Surface area = 2 (lb + bh + hl)

Lets assume TSA to be A

=> A = 2 (lb + bh + hl)

=> A  = 2lb + 2bh + 2hl

=> A = 2lb + 2h (l + b)

=> A – 2lb = 2h (l + b)

=> (A – 2lb) / 2 (l + b) = h

=> h = (A – 2lb) / 2 (l + b)

This is the formula for height when TSA is given.

Sample Problem 

Question 1: Calculate the height of a cuboid whose volume is 275 cm³ and its base area is 25 cm².

Solution: 

Volume of cuboid (V) = l × b × h

Given, Base area = l × b = 25 cm²

Hence,

V = (l × b) × h

275 = 25 × h

h = 275/25 = 11 cm

Therefore height of cuboid is 11 cm.

Question 2: The length, width, and height of a cuboidal shape box are 20 cm, 15 cm, and 10 cm, respectively. Find the total surface area of the box.

Solution:

 Total surface area of cuboid = 2(lb + bh + hl)

TSA = 2 (20 × 15 +15 × 10 + 10 × 20)

TSA = 2 (300 + 150 + 200) = 1300 cm²

Question 3: If a cube has a side length equal to 6 cm, then calculate its area.

Solution:

Given, side of cube = 5cm

Area of cube = 6 × side² = 6 × 5 × 5 = 150 sq.cm

Question 4: Calculate the height of the cylinder whose radius is 7 cm and the total surface area is 968 cm².

Solution: 

Let height of the cylinder = h, radius = r = 7cm

Total surface area of cylinder = 2πr (h + r)

TSA of a cylinder = 2 × (22/7) × 7 × (7 + h) = 968

Height of cylinder = 15 cm

Question 5: If a field is in the form of the rectangle if fencing of field is to be done and area of the field is 108m². One of its lengths is 12m and calculate the cost of fencing if the cost is Rs. 2 per meter.

Solution

Given,

Area of field = 108 m²

Length of field = 12m

Area of field = l x b

108  = 12  x b

b = 108/12 = 9m

perimeter of field = 2 × (l + b) = 2 × (12 + 9) = 2 × 21 = 42 m

Cost of fencing = 42 × 2 = Rs. 84

Question 6: Calculate the area of an equilateral triangle, if its side is 4cm.

Solution

Area of  equilateral triangle = √3/4 × side²

= √3/4 × 4 × 4

= 4√3 cm²

Question 7: Calculate the volume of the sphere if its radius is 7cm.

Solution:

Volume of sphere = 4/3 π × radius³ cubic units.

= 4/3 × 22/7 × 7³

= 88/3 × 7 × 7

= 1437.33 cm³