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Learning Resources

Area

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 08 Pathway: N/A

First Principles

Objective: Calculate the total surface area of composite shapes made by joining or cutting out rectangles, triangles, circles, and semicircles.

Interactive Composite Area Explorer

Choose a shape type and adjust dimensions to see how areas decompose!

(a) Concrete Scenario: A school in Eldoret is constructing a sports field shaped like a standard stadium running track—a central rectangular football pitch bounded by two semicircular curves. To calculate the total cost of planting Kikuyu grass, the groundskeeper breaks the shape down into standard geometric regions.

(b) Geometric Insight: Any complex 2D region can be analyzed by either additive composition (adding simple component areas together) or subtractive composition (subtracting circular cut-outs or holes from a outer boundary region).

(c) Algebraic Rule: For additive composite shapes: \[A_{\text{total}} = A_{\text{rectangle}} + A_{\text{semicircle}} = (l \times w) + \frac{1}{2}\pi r^2\] For subtractive composite shapes: \[A_{\text{remaining}} = A_{\text{outer}} - A_{\text{hole}} = (l \times w) - \pi r^2\]

Key Tip: Always check whether a curved part represents a full circle (\(\pi r^2\)), a semicircle (\(\frac{1}{2}\pi r^2\)), or a quarter-circle (\(\frac{1}{4}\pi r^2\)).

Key Formulas

\(A_{\text{rect}} = l \times w\) — Area of a rectangle where \(l\) is length and \(w\) is width.
\(A_{\text{circle}} = \pi r^2\) — Area of a complete circle with radius \(r = \frac{d}{2}\).
\(A_{\text{semicircle}} = \frac{1}{2}\pi r^2\) — Area of half a circle.
\(A_{\text{quarter}} = \frac{1}{4}\pi r^2\) — Area of a quarter circle.
\(A_{\text{additive}} = A_1 + A_2 + \dots + A_n\) — Total area of joined components.
\(A_{\text{subtractive}} = A_{\text{outer}} - A_{\text{inner hole}}\) — Remaining area after removing a region.

Worked Examples

Example 1 (Easy - Rectangle with Semicircular End): A wooden signboard has a rectangular base \(14\text{ cm}\) wide and \(20\text{ cm}\) tall, topped by a semicircle across its top width of \(14\text{ cm}\). Find the total area of the signboard. (Use \(\pi = \frac{22}{7}\)).
  1. Identify component shapes: One rectangle (\(20\text{ cm} \times 14\text{ cm}\)) and one semicircle with diameter \(d = 14\text{ cm}\) (hence radius \(r = 7\text{ cm}\)).
  2. Calculate rectangle area: \(A_{\text{rect}} = 20 \times 14 = 280\text{ cm}^2\).
  3. Calculate semicircle area: \(A_{\text{semi}} = \frac{1}{2} \times \frac{22}{7} \times 7^2 = \frac{1}{2} \times \frac{22}{7} \times 49 = 77\text{ cm}^2\).
  4. Combine areas: \(A_{\text{total}} = 280 + 77 = 357\text{ cm}^2\).
  5. Answer: \(357\text{ cm}^2\).
Example 2 (Medium - Rectangular Plate with Circular Hole): A metal sheet measuring \(30\text{ cm}\) by \(20\text{ cm}\) has a circular hole of radius \(7\text{ cm}\) punched through its center. Calculate the area of the remaining metal. (Use \(\pi = \frac{22}{7}\)).
  1. Identify components: Subtractive composition — Outer rectangle minus inner circle.
  2. Calculate rectangle area: \(A_{\text{rect}} = 30 \times 20 = 600\text{ cm}^2\).
  3. Calculate circular hole area: \(A_{\text{circle}} = \frac{22}{7} \times 7^2 = 154\text{ cm}^2\).
  4. Subtract hole area: \(A_{\text{remaining}} = 600 - 154 = 446\text{ cm}^2\).
  5. Answer: \(446\text{ cm}^2\).
Example 3 (Hard - Nyayo Stadium Running Track Infield): A athletic field consists of a central rectangle \(100\text{ m}\) long and \(56\text{ m}\) wide, flanked by two semicircular sections on both ends (each having a diameter of \(56\text{ m}\)). Find the total grass area of the entire infield in \(\text{m}^2\). (Use \(\pi = \frac{22}{7}\)).
  1. Identify components: Central rectangle plus two semicircles of radius \(r = \frac{56}{2} = 28\text{ m}\). Note that two semicircles equal one full circle!
  2. Calculate rectangle area: \(A_{\text{rect}} = 100 \times 56 = 5,600\text{ m}^2\).
  3. Calculate combined curved area (1 full circle): \(A_{\text{circle}} = \frac{22}{7} \times 28^2 = \frac{22}{7} \times 784 = 2,464\text{ m}^2\).
  4. Total field area: \(A_{\text{total}} = 5,600 + 2,464 = 8,064\text{ m}^2\).
  5. Answer: \(8,064\text{ m}^2\).

Common Mistakes

Mistake Using the diameter instead of the radius in circular area formulas.
Correction Always halve the given diameter first (\(r = \frac{d}{2}\)) before squaring! Squaring a diameter gives four times the actual area.
Why it feels right Textbooks often state the full width across a semicircle, which makes it tempting to plug that number straight into \(\pi r^2\).
Mistake Adding perimeter boundaries together when trying to find composite area.
Correction Area measures 2D space inside, while perimeter measures the outline. Deconstruct the shape into separate areas, compute each independently, and sum them.
Why it feels right Students confuse boundary length addition with surface region addition.

Real World

Stadium Turf Allocation: Multi-purpose stadiums like Kasarani combine rectangular soccer fields with curved athletic track end zones. Engineers use composite circular area formulas to order sod grass rolls.
Water Reservoir Construction: Concrete cattle dips and village water pans in Kajiado are frequently constructed with rectangular bodies and semicircular end vaults to optimize structural pressure and holding volume.
Custom Metal & Glass Fabrication: Artisans in Gikomba creating arched steel window grilles and fanlight doors calculate total glass area by combining rectangular lower panes with semicircular upper archways.

Practice

A metal rectangular badge measures 10 cm long and 6 cm wide. A semicircular crest of diameter 6 cm is attached to its top edge. What is the total area of the badge in square centimetres? (Use \u03C0 = 3.14) (Type only the number, e.g., 74.13)
Review the concepts above.
A wooden tabletop is shaped like a square with a side length of 14 cm, topped with a semicircle along one side (diameter = 14 cm). Find the total surface area of the tabletop in cm\u00B2. (Use \u03C0 = 22/7) (Type only the number, e.g., 273)
Review the concepts above.
A rectangular aluminum plate measures 20 cm by 15 cm. A circular hole of radius 7 cm is punched out of the center. What is the remaining area of the aluminum plate in cm\u00B2? (Use \u03C0 = 22/7) (Type only the number, e.g., 146)
Review the concepts above.
A square garden plot with sides of 20 metres has four quarter-circle flower beds (each of radius 7 metres) planted at its four corners. What is the remaining area of the central lawn in square metres? (Use \u03C0 = 22/7) (Type only the number, e.g., 246)
Review the concepts above.
A school running track field in Kisumu has a central rectangular field 80 metres long and 42 metres wide, with two semicircular ends across its 42 m width. What is the total area enclosed by the track in square metres? (Use \u03C0 = 22/7) (Type only the number, e.g., 4746)
Review the concepts above.
A church arched glass window consists of a rectangle 2 metres wide and 3 metres high, topped by a semicircle of diameter 2 metres. Calculate the total glass surface area in square metres. (Use \u03C0 = 3.14, round answer to 2 decimal places) (Type only the number, e.g., 7.57)
Review the concepts above.