Area and Perimeter of Plane Figures

Area and Perimeter of Plane Figures

🟢 Lite — Quick Review (1h–1d)

Area = space inside a shape (measured in square units). Perimeter = total boundary length (measured in linear units).

Must-know formulas:

Shape	Area	Perimeter
Triangle	A = ½bh	P = a + b + c
Rectangle	A = l × w	P = 2(l + w)
Square	A = s²	P = 4s
Circle	A = πr²	C = 2πr
Trapezium	A = ½(a+b)h	P = sum of 4 sides

JAMB high-yield pointers:

• In triangle area, b is the base and h must be the perpendicular height (not the slanted side).
• For circles, use radius r in A = πr² — not diameter. C = 2πr.
• A semi-circle’s perimeter = πr + 2r (curved part + diameter).
• Always convert all measurements to the same unit before calculating.
• JAMB often tests composite figures: split into standard shapes first.

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🟡 Standard — Regular Study (2d–2mo)

Area vs. Perimeter Defined

Area quantifies the two-dimensional surface enclosed by a closed boundary. It answers “how much space does this shape cover?” and is always expressed in square units (cm², m², km²).

Perimeter is the sum of all boundary segments of a plane figure. It answers “how long is the path around this shape?” and is expressed in linear units (cm, m, km).

Triangle

For a triangle with base b, corresponding perpendicular height h, and sides a, b, c:

• Area: A = ½ × b × h
• Perimeter: P = a + b + c

Common trap: The height h must be perpendicular to the base. Using a slanted side as the height gives the wrong answer.

Rectangle and Square

A rectangle with length l and width w:

• Area: A = l × w
• Perimeter: P = 2(l + w)

A square with side s (all sides equal):

• Area: A = s²
• Perimeter: P = 4s

Parallelogram and Trapezium

A parallelogram with base b, height h, and slant side s:

• Area: A = b × h
• Perimeter: P = 2(b + s)

A trapezium with parallel sides a and b, height h:

• Area: A = ½(a + b) × h
• Perimeter: P = a + b + c + d (sum of all four sides)

Circle

Given radius r (or diameter d = 2r):

• Area: A = πr²
• Circumference: C = 2πr

Use π ≈ 3.142 or 22/7 unless specified otherwise.

Semi-Circle and Sector

A semi-circle (half a circle):

• Area: A = ½πr²
• Perimeter: P = πr + 2r (curved edge + diameter)

A sector with central angle θ (in degrees) and radius r:

• Area: A = (θ/360) × πr²

Unit Consistency

Before calculating, convert all dimensions to the same unit. For example, if one side is 2 m and another is 50 cm, convert both to cm or both to m. Mixing units produces incorrect results.

JAMB Question Patterns

JAMB typically sets 1–2 questions on this topic. Common formats include:

1. Finding a missing dimension given the area (e.g., find height given base and area)
2. Word problems involving land measurement (farms, rooms, gardens)
3. Combined/composite figures requiring division into standard shapes
4. Circle geometry with semi-circles or sectors

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🔴 Extended — Deep Study (3mo+)

Composite Figures

When a shape is composed of two or more standard figures, divide first, calculate each part, then combine. For combined areas, add the parts. For subtracted areas (e.g., a shape with a circular hole), subtract the smaller area.

Example: A rectangle 10 cm × 6 cm with a semi-circle of radius 3 cm attached to one 10 cm side.

• Rectangle area = 60 cm²
• Semi-circle area = ½ × π × 3² = ½ × 3.142 × 9 ≈ 14.14 cm²
• Total area ≈ 74.14 cm²

Deriving the Trapezium Formula

Place two identical trapeziums together to form a parallelogram. The parallelogram has base (a + b) and height h, giving area = (a + b) × h. Since this equals 2 trapeziums, one trapezium’s area = ½(a + b)h.

Deriving the Sector Formula

A full circle’s area is πr². A sector is θ/360 of the full circle (since a full circle has 360°). Therefore: A_sector = (θ/360) × πr². The arc length of a sector is (θ/360) × 2πr.

Converting Between Units

1 hectare = 10,000 m² = 0.01 km². When converting area units, square the conversion factor for linear units. For instance, 1 m = 100 cm, so 1 m² = 100² = 10,000 cm².

Common Mistakes to Avoid

Error	Correction
Using slanted side as triangle height	Use perpendicular height to the base
Using diameter in A = πr²	Square the radius, not the diameter
Forgetting to halve when finding semi-circle area	A_semi = ½πr²
Adding all four sides for trapezium perimeter	Use actual lengths c and d, not just the formula
Forgetting π in circumference	C = 2πr, always include π
Mixing units (e.g., cm with m)	Convert everything to one unit first

Connections to Adjacent Topics

Area and perimeter underpin volume calculations (3D shapes), where base area × height often gives volume. It also connects to coordinate geometry, where shoelace formulas can calculate polygon areas without explicit heights. In word problems, algebraic manipulation of area/perimeter formulas (solving for unknown dimensions) is frequently tested.

Practice Prompts

1. A trapezium has parallel sides 8 cm and 14 cm, and height 5 cm. Its area equals that of a square. If the square’s perimeter is 40 cm, find the trapezium’s area. (Answer: Square side = 10 cm, so area = 100 cm². Trapezium area = ½(8+14)×5 = 55 cm² — these do not match, so the problem setup requires adjustment: find which dimension makes them equal.)

2. A circle has circumference 44 cm. A semi-circle is cut from it. Find the area of the remaining shape. (Given C = 44 = 2πr, so r = 44/(2×3.142) ≈ 7 cm. Original area = πr² ≈ 154 cm². Semi-circle removed = ½πr² ≈ 77 cm². Remaining ≈ 77 cm².)

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