Plane Areas: Triangles, Quadrilaterals, Circles & Composite Shapes

TAKEAWAYS:

• Triangles: Area = (base × height) / 2. Special cases include right triangles, equilateral triangles, and isosceles triangles.
• Quadrilaterals: Area formulas for squares, rectangles, parallelograms, trapeziums, and rhombuses depend on their sides and heights.
• Circles: Area = πr². Learn to find the area of semicircles, quarter circles, and sectors using the angle at the center.
• Composite Shapes: Break complex shapes into simpler parts (triangles, rectangles, circles) and add or subtract their areas.
• Units: Always express answers in square units (cm², m², km²) and convert units when necessary.
• Applications: Use area calculations in real-life problems like tiling floors, painting walls, or designing gardens.

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1. Area of Triangles

A triangle is a three-sided polygon. The area of a triangle depends on its base (b) and height (h).

Formula

The general formula for the area of a triangle is: $$\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$$

Types of Triangles and Their Areas

1. Right-Angled Triangle
   • One angle is 90°.
   • The two shorter sides (legs) can act as base and height. figure {"type":"rectangle","length":"10 cm","breadth":"6 cm","caption":"Rectangle with length 10 cm and width 6 cm."}

```figure
{"type":"circle","center":"O","radius":"3 cm","chord":true,"angleAtCentre":"180°","caption":"Semicircle with radius 3 cm (diameter = 6 cm)."}

Solution:

1. Area of rectangle = $10\times 6=60{\text{ cm}}^2$.
2. Area of semicircle = $\frac{1}{2}\times \pi \times 3^2=14.13{\text{ cm}}^2$.
3. Total area = $60-14.13=45.87{\text{ cm}}^2$.

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5. Applications of Area

Area calculations are used in many real-life situations:

• Land Measurement: Farmers calculate the area of their fields.
• Construction: Builders determine how much paint or tiles are needed.
• Gardening: Gardeners design flower beds or lawns.
• Sports: Cricket pitches, football fields, and basketball courts have specific area requirements.

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Exam Tip

1. Memorize Formulas: Know the area formulas for all shapes (triangle, rectangle, circle, etc.).

2. Label Diagrams: Always draw and label diagrams clearly. This helps in visualizing the problem.

3. Unit Consistency: Ensure all measurements are in the same unit (cm, m) before calculating.

4. Break Down Composite Shapes: If a shape is complex, divide it into simpler parts.

5. Practice SEE-Style Questions: Solve past SEE exam questions to understand the pattern.

   • Example Question:

     A rectangular plot has a length of 20 m and a width of 15 m. A semicircular garden with a diameter of 10 m is built inside the plot. Find the area of the remaining part of the plot. Solution:

     1. Area of rectangle = $20\times 15=300{\text{ m}}^2$.
     2. Radius of semicircle = $\frac{10}{2}=5\text{ m}$.
     3. Area of semicircle = $\frac{1}{2}\times \pi \times 5^2=39.25{\text{ m}}^2$.
     4. Remaining area = $300-39.25=260.75{\text{ m}}^2$.

6. Check Units: Always express the final answer in square units (cm², m², etc.).

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Practice Questions for SEE Exam

1. Find the area of a triangle with base 12 cm and height 8 cm.
2. A rectangle has a length of 15 cm and a width of 10 cm. Find its area.
3. Calculate the area of a circle with radius 7 cm (use $\pi =\frac{22}{7}$).
4. Find the area of a trapezium with parallel sides 6 cm and 10 cm, and height 4 cm.
5. A square has a side of 9 cm. A semicircle is drawn on one of its sides. Find the total area of the shape.
6. The area of a rhombus is 48 cm². If one of its diagonals is 8 cm, find the other diagonal.
7. A composite shape consists of a rectangle (length 12 cm, width 5 cm) with a semicircle (radius 3 cm) attached to one of its sides. Find the total area.
8. The area of an equilateral triangle is 25√3 cm². Find the length of its side.

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Answers:

1. 48 cm²
2. 150 cm²
3. 154 cm²
4. 32 cm²
5. 81 + 35.5 = 116.5 cm²
6. 12 cm
7. 60 + 14.13 = 74.13 cm²
8. 10 cm