Perimeter and Area of Plane Shapes

Perimeter and Area of Plane Shapes

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Perimeter is the total length around the boundary of a plane shape (a linear measurement in cm, m, etc.), while area is the size of the surface enclosed by that shape (a two-dimensional measurement in cm², m², etc.). For NCEE, the six shapes you must master are the rectangle, square, triangle, parallelogram, trapezium, and circle.

Must-know formulas:

• Rectangle: P = 2(l + w), A = l × w
• Square: P = 4s, A = s²
• Triangle: A = ½ × b × h
• Parallelogram: A = b × h
• Trapezium: A = ½(a + b)h
• Circle: C = 2πr, A = πr² (use π = 22/7)

High-yield pointers: (1) All lengths must be in the same unit before calculating. (2) Always write area with squared units. (3) The perpendicular height, not the slant side, is used in triangle and parallelogram area formulas.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Definitions

Perimeter (P) is the sum of the lengths of all sides of a closed plane figure. Because it measures only length, it is expressed in linear units: mm, cm, m, or km. Area (A) is the quantity of surface enclosed by the figure, expressed in square units (mm², cm², m², km²).

Rectangles and Squares

A rectangle with length l and width w has P = 2(l + w) and A = l × w. A square is a special rectangle where all four sides are equal (s), giving P = 4s and A = s². For example, a square with s = 6 cm has P = 24 cm and A = 36 cm².

Triangles and Parallelograms

The area of any triangle is A = ½ × b × h, where b is the base and h is the perpendicular height (not the slant length). A parallelogram’s area is A = b × h — the same product as a rectangle, which is why a parallelogram can be re-arranged into a rectangle of equal area. Both require perpendicular height, never the slanted side.

Trapezium

A trapezium has one pair of parallel sides a and b, plus perpendicular height h. Its area is the average of the parallel sides multiplied by the height: A = ½(a + b)h. The perimeter is the sum of all four sides.

Circles

The distance around a circle is the circumference, C = 2πr = πd, where r is the radius and d is the diameter (d = 2r). The area enclosed is A = πr². In NCEE, use π = 22/7 whenever the radius or diameter is a multiple of 7; otherwise use π = 3.142.

Unit Conversions

1 m = 100 cm, so 1 m² = 10,000 cm². Always convert before substituting into formulas.

Typical NCEE Question Patterns

• Compute perimeter or area given side lengths.
• Find a missing side given the total perimeter or area.
• Convert between cm² and m².
• Apply 22/7 for circle problems.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Composite Figures

NCEE often combines two or more basic shapes into one figure (an L-shape, a rectangle with a semicircle removed, a square with a triangle on top). To solve these, decompose the figure into rectangles, triangles, trapeziums, or sectors, then add (or subtract) the individual areas. Sketch and label every part before substituting values.

Same Perimeter, Different Areas — A Key Insight

Among rectangles with a fixed perimeter, the square encloses the largest area. For example, perimeter 24 cm can form a 6×6 square (A = 36 cm²) or a 9×3 rectangle (A = 27 cm²). This inverse relationship between regularity and area is tested as a conceptual MCQ in NCEE.

Edge Cases in Circles

Watch for these traps: (i) the question gives the diameter, but the formula needs the radius — divide by 2 first; (ii) a semicircle’s perimeter is πr + 2r (curved part + diameter), not πr; (iii) a quarter-circle’s perimeter is πr/2 + 2r. A circular ring’s area is π(R² − r²).

Common Mistakes

• Writing area units as cm instead of cm².
• Using diameter in πr².
• Mixing cm and m without conversion (causes a 10,000× error in area).
• Computing only three sides for a quadrilateral’s perimeter.
• Using the slant side instead of the perpendicular height for triangles and parallelograms.
• Substituting 22/7 for π when 3.142 is more accurate (or vice versa).

Worked Mini-Example

Find the area of a trapezium with parallel sides 8 cm and 12 cm and perpendicular height 5 cm. A = ½(a + b)h = ½(8 + 12)(5) = ½ × 20 × 5 = 50 cm².

Practice Prompts

1. A rectangular field is 45 m long and 30 m wide. Find its perimeter and the cost of fencing it at ₦2,500 per metre.
2. A circular garden has radius 14 m. Using π = 22/7, find (a) its circumference and (b) the area. Then convert the area into cm².

Exam Strategy for NCEE

Perimeter and area carry roughly 4% of the NCEE Mathematics paper — usually 1–2 objective questions. Budget about 90 seconds per question. Memorise the six formulas, the π values, and the unit conversions; the rest is careful substitution.

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