Mensuration Areas Perimeters Class 10 Questions for UP POLICE CONSTABLE

Rectangle and Square

These are the foundation shapes. A rectangle has length and width; a square has all sides equal. In UP Police exams, rectangle questions often involve fencing a field (perimeter) or painting a wall (area). Square problems test whether you remember that diagonal = side × root 2, which sometimes appears in indirect questions.

• Rectangle: Perimeter = 2(L + B), Area = L × B
• Square: Perimeter = 4 × side, Area = side²
• Square diagonal = side × √2 (useful for indirect problems)
• Common trap: confusing area and perimeter units (cm² vs cm)
• Real-world: fencing cost = perimeter × rate per unit

Triangle

Triangle questions in UP Police usually involve right triangles or equilateral triangles. The most common formula is half × base × height. For right triangles, you may need Pythagoras. Equilateral triangle area uses the formula with √3, which appears occasionally.

• General triangle: Area = (1/2) × base × height
• Right triangle: Area = (1/2) × leg1 × leg2
• Equilateral triangle: Area = (√3/4) × side²
• Perimeter = sum of all three sides
• Heron's formula rarely asked in UP Police (skip unless time permits)

Circle

Circle questions test radius vs diameter confusion and whether you use π correctly. UP Police papers often ask about area of a circular field, circumference for a circular fence, or comparing two circles. Remember: radius is half diameter. Use π = 22/7 for exact answers, π ≈ 3.14 for approximations.

• Circumference (perimeter) = 2πr or πd
• Area = πr²
• Radius = diameter/2
• Common mistake: using diameter in area formula instead of radius
• Sector area = (θ/360) × πr² (less common but appears sometimes)

Combined Shapes and Application Problems

UP Police loves mixing shapes. A question might ask for area of a rectangle with a semicircle on top (like a door), or a square with a circle inside. The key is to break the shape into known pieces, calculate each, then add or subtract. Always draw a rough sketch to avoid missing parts.

• Identify each simple shape within the complex shape
• Calculate area/perimeter of each part separately
• Add areas if shapes are joined; subtract if one is cut out
• Watch units: convert all to same unit before calculating
• Sketch the problem—visual clarity prevents errors

Perimeter vs Area: Common Confusion

Aspirants often mix up perimeter (distance around) and area (space inside). Perimeter is measured in units (m, cm); area is measured in square units (m², cm²). A small perimeter can have large area and vice versa. In UP Police, questions test this by asking 'how much fencing' (perimeter) vs 'how much paint' (area).

• Perimeter = boundary length (units: m, cm, km)
• Area = space covered (units: m², cm², hectare)
• A long thin rectangle can have same area as a square but different perimeter
• Fencing, border, fence = perimeter questions
• Paint, tile, carpet, grass = area questions

• Using diameter instead of radius in circle area formula. Always check: is it radius or diameter given? If diameter, divide by 2 first.
• Forgetting to square the radius. Area = πr² not πr. This is the #1 mistake in UP Police papers.
• Mixing units. If length is in meters and width in centimeters, convert to same unit before multiplying.
• In combined shapes, forgetting to subtract. If a circle is cut out from a square, you must subtract circle area from square area.
• Using wrong value of π. Use 22/7 for exact answers; use 3.14 only if problem says 'approximately' or gives π = 3.14.
• Reading 'perimeter' as 'area' or vice versa. Slow down and read the question twice.

• LBSCT = Length, Breadth, Side, Circle, Triangle. These are the 5 main shapes in UP Police mensuration.
• Area = INSIDE (space), Perimeter = OUTSIDE (boundary). Think of perimeter as the fence around a field.
• For circles: 2πr for circumference (2 in front), πr² for area (squared). The squared is the key difference.
• Equilateral triangle has √3 in formula because all sides equal—remember √3 ≈ 1.73, so area is roughly 0.43 × side².
• Sector = slice of pie. Angle/360 tells you what fraction of the full circle you have.

• UP Police papers typically have 1 direct formula question (easy, 30 seconds) and 1-2 application questions (medium, 2-3 minutes). Prioritize speed on direct questions.
• Recent papers show a trend toward combined shapes (rectangle + semicircle, square with circle inside). Practice these combinations.
• Always draw a rough diagram in the margin. This prevents misreading and helps you spot if you need to add or subtract areas.
• Use π = 22/7 unless told otherwise. This avoids decimal confusion and is faster in mental math.
• Watch for 'cost' or 'rate' questions. These require perimeter or area first, then multiply by given rate. Two-step problems are common.
• If a question seems too easy, re-read it. UP Police often hides a twist like 'both sides' or 'minus the door' that changes the calculation.