How to Convert One Shape to Another Using Area Formula and Step by Step Examples
Solids are of varying sizes and we come across them in our day to day life. Each solid that has been cast into a particular shape might not be in that shape earlier. Let’s take jewellery making for instance. How is it possible that perfectly round and circular pearls come out of the earth? It is only possible after converting a particular mould into a desired shape or size. When we convert any shape into any other shape, we are interested in the total volume of the solid. The total volume of the shape remains the same (assuming that there is no wastage in conversion). Generally, we heat the solid to convert it into the desired shape.
Let us study the conversion of solid from one shape to another in detail. We will cover the important underlying aspects of conversion. We will discuss the mathematical aspects of the conversion of solids. We will study the formulas of conversion from one shape to another.
Conversion of Solid From One Shape to Another
The space a solid occupies or contains is defined as the volume. Two solids of different shape can have the same volume. A solid, if converted into a different shape, will keep its original volume, no matter the shape in which it is converted.
For example, if we melt one big cylindrical candle into ten smaller candles, the total volume of the candles will be equal to the larger candle.
Even if the conversion is to a different shape, the volume remains the same. If we melt a big iron cube into smaller balls of different radiuses. The total volume of the solid remains the same.
Let’s understand it with the help of an example.
An iron ball of radius 21 cm is melted into 27 smaller balls of radius 7cm.
Let’s prove this statement using the concepts of the volume of the sphere.
The volume of the sphere is given by
\[Volume = \frac{4}{3} \pi r^{3}\]
The volume of the iron ball of radius 21cm
\[V = \frac{4}{3} \times \frac{22}{7} \times 21 \times 21 \times 21\]
\[V = 4 \times 22 \times 21 \times 21\]
\[V = 38, 808 cm^{3}\]
The volume of 27 iron ball of radius 7 cm
\[V = \frac{4}{3} \times \frac{22}{7} \times 7 \times 7 \times 7\]
\[V = 4 \times 22 \times 9 \times 7 \times 7\]
\[V = 38, 808 cm^{3}\]
The total volume of 27 balls of radius = 7cm is equal to the volume of the iron ball of radius = 21cm.
Conversion From One Shape to Another
When we convert a solid from one shape to another, the volume of the solid remains the same. Given below is the list of formulas pertaining to different shapes regarding their size and volumes.
In case of a cylinder of height h and radius r.
Volume = πr2h
Total surface area = 2πr(h + r)
In case of a cuboid of length l, height h and breadth b.
Volume = l * b * h
Total surface area = 2(lb+bh+ha)
In case of a cube of side a.
Volume = a3
Total surface area = 6a2
In case of a sphere of radius r.
Volume = 4/3r3
Total surface area = 4r2
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Problem 1: Rainwater is collected in a flat surface of length 6m and breadth 4m. It is then transferred into a cylindrical vessel of radius 10cm. What will be the height up to which the vessel will be filled if 2cm rainfall has fallen?
Problem 2: The external dimensions of a cuboid are 20cm x 30cm x 25 cm. The width of the wood used to make the cuboid is 1.5cm. Find the amount of wood used in making the cuboidal shape.
Problem 3: A cinema hall wants to make preparations for attendance to maintain social distancing. Normally, each person requires 150cm3 of air but under new guidelines, each person needs to be given 600cm3 of space. The dimensions of the hall are 100m x 60m x 15m. Find the number of people that can sit in the hall before and after the restriction.
Problem 4: Find the total volume of 3 spheres with diameters 14cm, 21cm and 42cm. Also, these three spheres are combined to make a single sphere. Find the diameter of the sphere thus formed.
FAQs on Conversion of One Shape to Another in Mensuration
1. What does conversion of one shape to another mean in maths?
Conversion of one shape to another means changing a given shape into a different shape without changing its area. In mensuration, this process keeps the area constant while the dimensions change.
- The original and new shapes have equal areas.
- Only the dimensions (length, breadth, radius, etc.) are recalculated.
- It is commonly used in geometry problems involving area comparison.
2. How do you convert a rectangle into a square of equal area?
To convert a rectangle into a square of equal area, first find the rectangle’s area and then calculate the side of the square using that area.
- Area of rectangle = length × breadth
- Let this area = A
- Side of square = √A
3. How do you convert a square into a rectangle of equal area?
To convert a square into a rectangle of equal area, keep the area constant and adjust the length and breadth accordingly.
- Area of square = side²
- Choose a new length
- Find breadth = Area ÷ length
4. How do you convert a rectangle into a circle of equal area?
To convert a rectangle into a circle of equal area, equate the rectangle’s area to the circle’s area formula and solve for the radius.
- Area of rectangle = l × b
- Area of circle = πr²
- Set l × b = πr² and solve for r
5. What is the formula used when converting one 2D shape to another?
The main rule for converting one 2D shape to another is to equate their area formulas.
- Rectangle: l × b
- Square: side²
- Circle: πr²
- Triangle: ½ × base × height
6. Can you give an example of converting a triangle into a rectangle of equal area?
To convert a triangle into a rectangle of equal area, first calculate the triangle’s area and then assign suitable rectangle dimensions.
- Area of triangle = ½ × base × height
7. Why does the area remain the same when converting one shape to another?
The area remains the same because shape conversion problems are based on the principle of equal area transformation.
- The total surface covered does not change.
- Only the dimensions are adjusted.
- This ensures fair comparison between different geometric shapes.
8. How do you convert a circle into a square of equal area?
To convert a circle into a square of equal area, equate the circle’s area to the square’s area and solve for the side.
- Area of circle = πr²
- Area of square = side²
- So, side = √(πr²)
9. What are common mistakes when converting one shape to another?
The most common mistake in converting one shape to another is not equating the correct area formulas properly.
- Forgetting to use the correct formula (e.g., πr² for circle).
- Not squaring or square-rooting correctly.
- Using incorrect value of π (22/7 or 3.14).
- Changing dimensions without maintaining equal area.
10. Where is conversion of one shape to another used in real life?
Conversion of one shape to another is used in real life when redesigning spaces while keeping the same area.
- Changing rectangular land into circular parks.
- Redesigning floor layouts.
- Manufacturing materials into different shapes.
- Architecture and construction planning.
