Area and Perimeter of Shapes

1. The Core Concepts: Perimeter and Area

Before we dive into formulas, let's ensure we have a solid grasp of the basic concepts. Perimeter is the total distance around the boundary of a two-dimensional shape, while area is the measure of the space enclosed within that boundary.

Now, let's establish the foundational formulas for the simplest shapes.

Squares and Rectangles

• Rectangle:
  • Perimeter $P=2\times (\text{length}+\text{breadth})=2(l+b)$
  • Area $A=\text{length}\times \text{breadth}=l\times b$
• Square: (where side = $s$)
  • Perimeter $P=4\times \text{side}=4s$
  • Area $A=\text{side}\times \text{side}=s^2$

Circles

The perimeter of a circle has a special name: the circumference.

• Key terms:
  • Radius ($r$): The distance from the center to any point on the circle.
  • Diameter ($d$): The distance across the circle through its center. Remember that $d=2r$.
• Formulas:
  • Circumference $C=2\pi r$ or $C=\pi d$
  • Area $A=\pi r^2$
• For the CAT exam, you can usually approximate $\pi$ as $\frac{22}{7}$ or 3.14, depending on which is more convenient for the numbers in the problem.

2. Formulas for Triangles and Quadrilaterals

Now we'll expand our toolkit to include more complex shapes. The image below is an excellent reference sheet that summarizes the formulas for most of the shapes you'll encounter.

Let's discuss these in more detail.

Triangles

There are a few ways to calculate the area of a triangle, and choosing the right one is key to solving problems quickly.

1. Standard Formula (Base and Height): This is the most common formula.

   $$\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$$

   The 'height' must be perpendicular to the 'base'.

2. Heron's Formula (Three Sides): Use this when you know the lengths of all three sides ($a,b,c$) but not the height.

   • First, calculate the semi-perimeter, $s=\frac{a+b+c}{2}$.
   • Then, apply the formula: $$\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}$$

3. Equilateral Triangle (Side 'a'): For an equilateral triangle, you can use a direct shortcut formula:

   $$\text{Area}=\frac{\sqrt{3}}{4}{a}^2$$

Quadrilaterals

Beyond squares and rectangles, you need to know the formulas for these shapes.

Here is a summary of the key formulas:

• Parallelogram: Area = base × perpendicular height. Be careful not to use the slant height.
• Rhombus: Can be calculated in two ways:
  1. Area = base × perpendicular height (since it's a type of parallelogram).
  2. Area = $\frac{1}{2}\times d_1\times d_2$ (half the product of its diagonals). This is more commonly used in problems.
• Trapezium (or Trapezoid): Area = $\frac{1}{2}\times (\text{sum of parallel sides})\times \text{height}$.
• Kite: Area = $\frac{1}{2}\times d_1\times d_2$ (same as a rhombus).

3. Application and Strategy for CAT Problems

Memorizing formulas is the first step. The real challenge in the CAT is applying them correctly, often by combining concepts. Here is a simple strategy:

1. Identify the shape(s) involved in the problem. Is it a single shape or a composite one (e.g., a square with a circle cut out)?
2. List the knowns and unknowns. What information is given (sides, diagonals, radius)? What do you need to find (area, perimeter)?
3. Select the right formula. Based on your list, which formula connects the knowns to the unknown?
4. Check for intermediate steps. Do you need to use another concept (like the Pythagorean theorem from a previous lesson) to find a missing piece of information (like a height or diagonal)?

Let's walk through a few examples from a typical CAT question bank.

Example 1: Using Heron's Formula (Q20)

Problem: A triangle has sides 13 cm, 14 cm, and 15 cm. What is its area?

• Step 1 (Shape): A triangle.
• Step 2 (Knowns/Unknowns): Knowns are the three sides: $a=13,b=14,c=15$. Unknown is the area.
• Step 3 (Formula): Since we only have the sides, Heron's formula is the perfect tool.
• Step 4 (Calculation):
  • Semi-perimeter $s=\frac{13+14+15}{2}=\frac{42}{2}=21$.
  • Area = $\sqrt{s(s-a)(s-b)(s-c)}$
  • Area = $\sqrt{21(21-13)(21-14)(21-15)}$
  • Area = $\sqrt{21\times 8\times 7\times 6}$
  • To simplify the square root, break down the numbers into prime factors: $\sqrt{(3\times 7)\times (2^3)\times 7\times (2\times 3)}$
  • Group the factors: $\sqrt{2^4\times 3^2\times 7^2}=2^2\times 3\times 7=4\times 21=84$ cm².

Example 2: Working with Composite Shapes (Q27)

Problem: A rectangular field is 60 m long and 40 m wide. A path of 2 m is constructed inside along all sides. Find the area of the path.

• Step 1 (Shape): A large rectangle with a smaller rectangle inside. The area of the path is the difference between their areas.
• Step 2 (Knowns/Unknowns):
  • Outer rectangle: length = 60 m, width = 40 m.
  • Path width = 2 m.
  • Inner rectangle dimensions need to be calculated.
• Step 3 (Formula): Area of rectangle = $l\times w$. Area of path = Area(outer) - Area(inner).
• Step 4 (Calculation):
  • Area(outer) = $60\times 40=2400$ m².
  • The path is 2m wide on all sides. So, the inner rectangle's length is $60-2-2=56$ m.
  • The inner rectangle's width is $40-2-2=36$ m.
  • Area(inner) = $56\times 36=2016$ m².
  • Area of path = $2400-2016=384$ m². (Note: The answer in the provided resource is 392 m², which seems to be a calculation error. Our method is correct.)

Conclusion

Today we have covered the essential formulas and strategies for calculating the area and perimeter of the most common 2D shapes. This brings our module on Plane Geometry to a close.

Key Takeaways:

• Master the Formulas: You should be able to instantly recall the area and perimeter/circumference formulas for squares, rectangles, circles, triangles (standard and Heron's), parallelograms, rhombuses, and trapeziums.
• Know When to Use What: The key to speed is choosing the right formula for the given information (e.g., using Heron's for three known sides).
• Break Down Complex Shapes: Many CAT problems involve composite figures. The strategy is almost always to find the area by adding or subtracting the areas of simpler, known shapes.

Preview of the Next Lesson:
We are now ready to move from two dimensions to three! Our next lesson will kick off the Mensuration and Coordinate Geometry module. We will build directly on today's concepts to learn how to calculate the surface area and volume of 3D solids like cubes, cylinders, and cones.