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.
The area of a square is 144 cm². What is the length of its diagonal?
Practice 2easy
A parallelogram has a base of 20 cm and a height of 15 cm. What is its area?
Practice 3easy
The perimeter of a rectangle is 56 cm. If its length is 18 cm, what is its breadth?
Practice 4easy
A trapezium has parallel sides of 12 cm and 18 cm, and a height of 10 cm. What is its area?
Practice 5easy
A rhombus has diagonals of length 18 cm and 24 cm. What is its area?
Practice 6easy
A rectangle has a length of 24 cm and a breadth of 16 cm. What is the perimeter of the rectangle?
Practice 7medium
In a quadrilateral PQRS, the diagonals PR and QS intersect at point O. If PO = 6 cm, OR = 9 cm, QO = 4 cm, and OS = 8 cm, what is the ratio of the area of triangle POQ to the area of triangle ROS?
Practice 8medium
A square and a rhombus have the same perimeter of 48 cm. If the rhombus has one angle of 60°, what is the difference between the area of the square and the area of the rhombus?
Practice 9medium
The diagonals of a rhombus are 16 cm and 12 cm. What is the perimeter of the rhombus?
Practice 10medium
A parallelogram has adjacent sides of 12 cm and 8 cm. If the perpendicular distance between the longer sides is 6 cm, what is the perpendicular distance between the shorter sides?
Practice 11hard
A parallelogram ABCD has sides AB = 13 cm and BC = 8 cm. The angle between them is 60°. A perpendicular is drawn from vertex C to side AB, meeting it at point E. What is the length of CE?
Practice 12hard
In a rhombus PQRS, the diagonals PR and QS intersect at O. If PR = 30 cm, QS = 16 cm, and a point M on side PQ is such that OM ⊥ PQ, then the length of OM is:
Practice 13hard
A quadrilateral PQRS has sides PQ = 13 cm, QR = 14 cm, RS = 15 cm, and SP = 12 cm. Using Brahmagupta's formula, if the quadrilateral is cyclic, its area is closest to:
Practice 14hard
A trapezium ABCD has parallel sides AB = 24 cm and CD = 16 cm. The perpendicular distance between them is 10 cm. A line parallel to both AB and CD divides the trapezium into two parts of equal area. At what distance from AB is this dividing line located?
Practice 15hard
In a kite ABCD with AB = AD = 10 cm and CB = CD = 8 cm, the diagonals AC and BD intersect at O such that BO = 6 cm. What is the length of diagonal AC?
Practice 16hard
In a rectangle ABCD, the diagonals AC and BD intersect at O. If AB = 20 cm, BC = 15 cm, and P is a point on diagonal AC such that OP = 2 cm, then the distance from P to vertex A is:
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