Area Perimeter 2Efe28

1. **Problem Statement:** Understand the concepts of perimeter and area for various 2D shapes and how to calculate them. 2. **Definitions:** - Perimeter is the distance around a closed shape, measured in units like cm or m. - Area is the space inside a shape, measured in square units like cm² or m². 3. **Unit Conversion:** - Use the mnemonic "King Henry Died Miserable Death Called Measles" for units: km, hm, dam, m, dm, cm, mm. - For length, moving right multiplies by 10, moving left divides by 10. - For area, each step multiplies or divides by 100. - Always convert all sides to the same unit before calculating. 4. **Squares and Rectangles:** - Square perimeter formula: $$P = 4s$$ where $$s$$ is the side length. - Square area formula: $$A = s^2$$. - Rectangle perimeter formula: $$P = 2l + 2b$$ where $$l$$ is length and $$b$$ is breadth. - Rectangle area formula: $$A = l \times b$$. 5. **Worked Example 1 (Rectangle):** - Given $$l = 10\,cm$$ and $$b = 6\,cm$$. - Calculate perimeter: $$P = 2 \times 10 + 2 \times 6 = 20 + 12 = 32\,cm$$. - Calculate area: $$A = 10 \times 6 = 60\,cm^2$$. 6. **Worked Example 2 (Square):** - Given side $$s = 7\,cm$$. - Calculate perimeter: $$P = 4 \times 7 = 28\,cm$$. - Calculate area: $$A = 7^2 = 49\,cm^2$$. 7. **Triangles:** - Perimeter formula: $$P = a + b + c$$ where $$a,b,c$$ are side lengths. - Area formula: $$A = \frac{1}{2} \times b \times h$$ where $$h$$ is the perpendicular height. - Important: $$h$$ is perpendicular to base, not the slanted side. 8. **Worked Example (Triangle):** - Sides: $$26\,cm$$, $$133\,cm$$, $$145\,cm$$. - Height $$h = 24\,cm$$. - Calculate perimeter: $$P = 26 + 133 + 145 = 304\,cm$$. - Calculate area: $$A = \frac{1}{2} \times 133 \times 24 = 1596\,cm^2$$. 9. **Circles and Semicircles:** - Radius $$r$$ and diameter $$d = 2r$$. - Circumference formula: $$C = 2\pi r$$. - Area formula: $$A = \pi r^2$$. - Use $$\pi \approx 3,142$$. 10. **Worked Example (Circle):** - Given $$r = 9\,cm$$. - Calculate circumference: $$C = 2 \times 3,142 \times 9 = 56,56\,cm$$. - Calculate area: $$A = 3,142 \times 9^2 = 254,50\,cm^2$$. 11. **Semicircle:** - Area: $$A = \frac{1}{2} \pi r^2$$. - Perimeter: $$C = \pi r + d$$. 12. **Worked Example (Semicircle):** - Given area $$50\pi\,cm^2$$. - Solve for $$x$$ where $$x$$ is radius: $$50\pi = \frac{1}{2} \pi x^2 \Rightarrow 50 = \frac{1}{2} x^2 \Rightarrow x^2 = 100 \Rightarrow x = 10\,cm$$. 13. **Compound Figures:** - Split complex shapes into rectangles, squares, triangles, semicircles. - Calculate each area separately, then add. 14. **Worked Example 1 (Triangle + Rectangle):** - Triangle area: $$\frac{1}{2} \times 16 \times 6 = 48\,cm^2$$. - Rectangle area: $$16 \times 12 = 192\,cm^2$$. - Total area: $$48 + 192 = 240\,cm^2$$. 15. **Worked Example 2 (L-shape):** - Split two ways, both total $$76\,cm^2$$. 16. **Common Mistakes:** - Forgetting unit conversion. - Using slanted side as height. - Confusing units (cm vs cm²). - Mixing radius and diameter. - Rounding too early. 17. **Practice Questions:** - (1) Square with perimeter $$64\,cm$$: find side and area. - (2) Rectangle with area $$60\,cm^2$$ and length $$12\,cm$$: find breadth and perimeter. - (3) Circle with radius $$7\,cm$$, $$\pi = 3,142$$: find circumference and area. 18. **Summary:** - Formulas highlighted in yellow. - Units in green. - Answers in green boxes. **Final answers for practice:** - (1) Side $$s = \frac{64}{4} = 16\,cm$$, Area $$= 16^2 = 256\,cm^2$$. - (2) Breadth $$b = \frac{60}{12} = 5\,cm$$, Perimeter $$P = 2 \times 12 + 2 \times 5 = 34\,cm$$. - (3) Circumference $$C = 2 \times 3,142 \times 7 = 43,988\,cm$$, Area $$A = 3,142 \times 7^2 = 153,938\,cm^2$$.