Quadrilaterals, Polygons and Area Ratios Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Quadrilaterals, Polygons and Area Ratios notes for CAT: 70 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Quadrilaterals, Polygons and Area Ratios

CAT QA Geometry

Quadrilaterals, Polygons and Area Ratios

A staircase from basic four-sided area to polygon-level pattern recognition.

chapter PYQs

①

📐 t1 — Trapeziums, Parallelograms and Rectangles

You master height-based area, parallel-side logic, trapezium similarity, rectangle slicing, and incircle traps in trapeziums.

6 own-course PYQs High-moderate weight Selected topic

②

◆ t2 — Rhombus and Diagonal Geometry

You learn how perpendicular diagonals, side length, and area interact.

3 own-course PYQs

③

⬡ t3 — Regular Polygons and Hexagons

You move from four-sided figures to symmetry, interior angles, diagonals, and standard polygon areas.

6 own-course PYQs

④

🧩 t4 — Area Ratios and Quadrilateral Constraints

You finish with boundary conditions, side constraints, and ratio-based reasoning.

1 own-course PYQ

End goal: identify the figure type, choose the right height or diagonal relation, and avoid wasting time on unnecessary construction.

Topic Hero: Trapeziums, Parallelograms and Rectangles

Geometry → Quadrilaterals → t1

Trapeziums, Parallelograms and Rectangles

One-line hook: when sides are parallel, area starts obeying simple height rules.

What you'll learn here

Area formulas for trapeziums, parallelograms, and rectangles.
How common height creates area ratios.
How extended sides of a trapezium create similar triangles.
How rectangle slicing produces arithmetic or geometric area patterns.
How incircle conditions work in a trapezium.

The Common Thread: Base × Height Thinking

Most area questions are height questions

▱

Parallelogram

One base, one perpendicular height.

▭

Rectangle

Length times breadth; height is already visible.

⏢

Trapezium

Average of parallel sides times height.

CAT habit: first mark the parallel sides, then ask: “What is the height?”

Area Formula Toolkit

Three formulas you will actually use

Rectangle:

A = l b

Parallelogram:

A = bh

Trapezium:

A = \frac{1}{2} (a + b) h

Here $a, b$ are the two parallel sides.

Quadrilaterals, Polygons and Area Ratios: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability NAT

A rhombus has integer side length and integer diagonals. Its area is numerically equal to 24 times its side length. What is the perimeter of this rhombus?

Correct Answer:

100

Step-by-Step Solution

Key idea: This combines rhombus metric relations with Diophantine constraints. The condition "Area = 24 × Side" links the product of diagonals to their Pythagorean combination.

Step 1: Set up equations.

Let side be $s$ . Area $A = 24 s$ .

Also $A = \frac{1}{2} d_{1} d_{2} \Rightarrow d_{1} d_{2} = 48 s$ .

Rhombus diagonal identity: $d_{1}^{2} + d_{2}^{2} = 4 s^{2}$ .

Step 2: Form equation for sum of diagonals.

$(d_{1} + d_{2})^{2} = d_{1}^{2} + d_{2}^{2} + 2 d_{1} d_{2} = 4 s^{2} + 2 (48 s) = 4 s^{2} + 96 s$ .

For $d_{1} + d_{2}$ to be integer, $4 s^{2} + 96 s$ must be a perfect square.

Let $4 (s^{2} + 24 s) = k^{2} \Rightarrow s^{2} + 24 s = m^{2}$ .

Step 3: Solve Diophantine equation.

Complete the square: $(s + 12)^{2} - 144 = m^{2}$ .

$(s + 12)^{2} - m^{2} = 144 \Rightarrow (s + 12 - m) (s + 12 + m) = 144$ .

Let factors be $u, v$ with $uv = 144, v > u$ . Since sum $(s + 12 - m) + (s + 12 + m) = 2 (s + 12)$ is even, $u, v$ must have same parity. Both must be even.

Even factor pairs of 144: $(2, 72), (4, 36), (6, 24), (8, 18), (12, 12)$ .

Step 4: Test candidates for valid geometry.

Pair (2,72): $2 (s + 12) = 74 \Rightarrow s = 25$ .
Pair (4,36): $2 (s + 12) = 40 \Rightarrow s = 8$ .
Pair (6,24): $2 (s + 12) = 30 \Rightarrow s = 3$ .
Pair (8,18): $2 (s + 12) = 26 \Rightarrow s = 1$ .

Check existence: For real diagonals, we need $d_{1}^{2} + d_{2}^{2} \geq 2 d_{1} d_{2} \Rightarrow 4 s^{2} \geq 96 s \Rightarrow s \geq 24$ .

Only $s = 25$ satisfies this. ( $s = 8, 3, 1$ yield imaginary diagonals).

Step 5: Calculate perimeter.

Valid side $s = 25$ . Perimeter = $4 s = 100$ .

(Verification: If $s = 25$ , $d_{1} d_{2} = 1200, d_{1}^{2} + d_{2}^{2} = 2500$ . Diagonals are 30 and 40. Integers. Valid.)

Answer: 100

Question 2 · Quantitative Ability MCQ

A trapezium $A B C D$ has $A B ∥ C D$ . A circle is inscribed in the trapezium, tangent to all four sides. If $A B = 18$ cm, $C D = 8$ cm, and the area of the trapezium is $156$ sq. cm, what is the length, in cm, of each non-parallel side?

A.
13
B.
15
C.
17
D.
20

Correct Answer:

Step-by-Step Solution

Key idea: This uses the Tangential Trapezium Property combined with the Right Triangle Construction for non-parallel sides. Crucially, it requires verifying that the implied isosceles configuration is geometrically consistent.

Step 1: Use Tangential Quadrilateral Property (Pitot Theorem).

For any quadrilateral with an inscribed circle, sums of opposite sides are equal.

$A B + C D = A D + B C$ .

Given $A B = 18, C D = 8 \Rightarrow A D + B C = 26$ .

Step 2: Determine Height.

Area = $\frac{1}{2} (A B + C D) \times h = 156$ .

$\frac{1}{2} (26) h = 156 \Rightarrow 13 h = 156 \Rightarrow h = 12$ .

Step 3: Analyze Non-Parallel Sides.

The question asks for "the length of EACH non-parallel side", implying $A D = B C$ .

If $A D = B C$ , then $2 A D = 26 \Rightarrow A D = 13$ .

Step 4: Verify Consistency (Crucial Step).

In an isosceles trapezium, the projection of the slant side onto the base is $\frac{A B - C D}{2} = \frac{18 - 8}{2} = 5$ .

By Pythagoras, required height for leg 13 and projection 5 is $1 3^{2} - 5^{2} = 169 - 25 = 144 = 12$ .

This matches the calculated height $h = 12$ exactly.

Thus, the isosceles configuration is valid and the side length is indeed 13.

Answer: A

Quadrilaterals, Polygons and Area Ratios Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Chapter Roadmap: Quadrilaterals, Polygons and Area Ratios

Quadrilaterals, Polygons and Area Ratios

📐 t1 — Trapeziums, Parallelograms and Rectangles

◆ t2 — Rhombus and Diagonal Geometry

⬡ t3 — Regular Polygons and Hexagons

🧩 t4 — Area Ratios and Quadrilateral Constraints

Topic Hero: Trapeziums, Parallelograms and Rectangles

Trapeziums, Parallelograms and Rectangles

What you'll learn here

The Common Thread: Base × Height Thinking

Most area questions are height questions

Parallelogram

Rectangle

Trapezium

Area Formula Toolkit

Three formulas you will actually use

Quadrilaterals, Polygons and Area Ratios: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

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