finding area when diagonals are given, calculating perimeter from given conditions, property-based questions, and coordinate geometry problems involving quadrilaterals
⚡Powerful Shortcut - Diagonal Rule
For any quadrilateral with perpendicular diagonals, Area = (1/2) × d1 × d2. This works for rhombus, kite, and square. This single formula can solve multiple question types quickly.
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Worked Example
Solve this step-by-step before moving on
1
Step 1
Find Area
Area = (1/2) × d1 × d2
Area = (1/2) × 16 × 12 = 96 cm²
2
Step 2
Find Side Length
Diagonals of rhombus bisect at right angles.
Half diagonals: 8 cm and 6 cm
Using Pythagoras: Side = √(8² + 6²) = √(64 + 36) = √100 = 10 cm
3
Step 3
Find Perimeter
Perimeter = 4 × side = 4 × 10 = 40 cm
Another Time-Saving Trick:
When a quadrilateral's vertices are given in coordinate geometry, use the shoelace formula: Area = (1/2)|x1(y2-y4) + x2(y3-y1) + x3(y4-y2) + x4(y1-y3)|. This eliminates the need for multiple calculations.
Common Mistake:
Students often confuse rhombus and square properties. Remember: Square has all properties of rhombus PLUS all angles are 90°. Rhombus doesn't necessarily have 90° angles. Also, many forget that in trapezium area formula, height is the perpendicular distance between parallel sides, not the slant side length.
A rectangle has length 24 cm and breadth 18 cm. What is the perimeter of the rectangle?
Practice 2easy
The area of a square is 256 cm². What is the length of its side?
Practice 3easy
A parallelogram has a base of 20 cm and a height of 15 cm. What is its area?
Practice 4easy
A rhombus has diagonals of length 16 cm and 12 cm. What is its area?
Practice 5easy
A trapezium has parallel sides of 10 cm and 14 cm, and a height of 8 cm. What is its area?
Practice 6easy
The diagonals of a rectangle are 10 cm each. If one side is 6 cm, what is the length of the adjacent side?
Practice 7medium
A parallelogram has a base of 24 cm and a height of 15 cm. If the base is increased by 25% and the height is decreased by 20%, what is the percentage change in the area of the parallelogram?
Practice 8medium
The diagonals of a rhombus are 16 cm and 12 cm. What is the perimeter of the rhombus?
Practice 9medium
In a rectangle, the length is 8 cm more than the width. If the perimeter is 56 cm, what is the area of the rectangle?
Practice 10hard
A parallelogram PQRS has sides PQ = 13 cm and QR = 8 cm. The angle between them is 60°. A perpendicular is drawn from R to side PQ, meeting it at point T. What is the length of RT?
Practice 11hard
In a kite PQRS, the diagonals PR and QS intersect at O. If PR = 24 cm, QS = 18 cm, and the diagonals intersect such that PO = 16 cm and QO = 10 cm, what is the area of the kite?
60-Second Revision — Quadrilaterals
Remember: All quadrilateral interior angles sum to 360°
Formula: Area with perpendicular diagonals = (1/2) × d1 × d2
Trap: Square is both rectangle and rhombus, but rhombus is not always rectangle
Quick check: Parallelogram opposite sides are parallel AND equal
Memory aid: Rectangle = right angles, Rhombus = equal sides