Area and Perimeter. Grade 8 Mathematics: Area and Perimeter of 2D Shapes Subtopics Navigator Introduction to Area and Perimeter Square Rectangle Triangle Parallelogram Shape Comparison Real-World Applications Cumulative Exercises Conclusion Lesson Objectives Understand the concepts of area and perimeter Calculate perimeter of squares, rectangles, triangles, and parallelograms Calculate area of squares, rectangles, triangles, and parallelograms Apply formulas to solve real-world problems Distinguish between area and perimeter in practical situations Introduction to Area and Perimeter Perimeter is the total distance around the outside of a shape. It's measured in linear units (cm, m, km). Area is the amount of space inside a shape. It's measured in square units (cm², m², km²). SQUARE Square 4 equal sides4 right angles RECTANGLE Rectangle Opposite sides equal4 right angles TRIANGLE Triangle 3 sides3 angles PARALLELOGRAM Parallelogram Opposite sides paralleland equal Real-World Examples: Perimeter: Fencing a garden, framing a picture, running track Area: Flooring a room, painting a wall, land measurement Understanding Basic Concepts What is the difference between area and perimeter? Why is area measured in square units? Give an example of when you would need to calculate perimeter in real life. Square - Area and Perimeter A square has all four sides equal in length and all four angles are 90°. Square Formulas Perimeter = [latex]4 times text{side}[/latex] or [latex]4s[/latex] Area = [latex]text{side} times text{side}[/latex] or [latex]s^2[/latex] Example 1: Finding Square Perimeter Find the perimeter of a square with side length 8 cm. Perimeter = [latex]4 times 8 = 32[/latex] cm Answer: 32 cm Example 2: Finding Square Area Find the area of a square with side length 12 m. Area = [latex]12 times 12 = 144[/latex] m² Answer: 144 m² Example 3: Finding Side from Area A square has area 81 cm². What is its side length? Area = [latex]s^2 = 81[/latex], so [latex]s = sqrt{81} = 9[/latex] cm Answer: 9 cm Exercises (Squares) Find the perimeter of a square with side 15 cm Calculate the area of a square with side 7.5 m A square has perimeter 48 cm. What is its area? A square has area 64 m². What is its perimeter? Find the side length of a square with area 121 cm² Rectangle - Area and Perimeter A rectangle has opposite sides equal in length and all four angles are 90°. Rectangle Formulas Perimeter = [latex]2 times (text{length} + text{width})[/latex] or [latex]2(l + w)[/latex] Area = [latex]text{length} times text{width}[/latex] or [latex]l times w[/latex] Understanding Rectangle Perimeter 1 A rectangle has 2 lengths and 2 widths 2 Perimeter = length + width + length + width 3 This simplifies to: 2 × length + 2 × width 4 Or more simply: 2 × (length + width) Example 4: Finding Rectangle Perimeter Find the perimeter of a rectangle with length 14 cm and width 8 cm. Perimeter = [latex]2 times (14 + 8) = 2 times 22 = 44[/latex] cm Answer: 44 cm Example 5: Finding Rectangle Area Find the area of a rectangle with length 25 m and width 12 m. Area = [latex]25 times 12 = 300[/latex] m² Answer: 300 m² Example 6: Finding Missing Dimension A rectangle has area 180 cm² and length 15 cm. Find its width. Area = length × width [latex]180 = 15 times w[/latex] [latex]w = 180 div 15 = 12[/latex] cm Answer: Width = 12 cm Exercises (Rectangles) Find the perimeter of a rectangle with length 18 cm and width 10 cm Calculate the area of a rectangle with length 32 m and width 15 m A rectangle has perimeter 60 cm and length 18 cm. Find its width A rectangle has area 240 cm² and width 12 cm. Find its length Find the area of a rectangle with perimeter 40 m and length 12 m Triangle - Area and Perimeter A triangle has three sides and three angles. The sum of its angles is always 180°. Triangle Formulas Perimeter = [latex]a + b + c[/latex] (sum of all three sides) Area = [latex]frac{1}{2} times text{base} times text{height}[/latex] or [latex]frac{1}{2}bh[/latex] Understanding Triangle Area 1 The base is any side of the triangle 2 The height is the perpendicular distance from the base to the opposite vertex 3 A triangle is half of a rectangle with the same base and height 4 Therefore: Area = ½ × base × height Example 7: Finding Triangle Perimeter Find the perimeter of a triangle with sides 12 cm, 15 cm, and 18 cm. Perimeter = [latex]12 + 15 + 18 = 45[/latex] cm Answer: 45 cm Example 8: Finding Triangle Area Find the area of a triangle with base 20 cm and height 8 cm. Area = [latex]frac{1}{2} times 20 times 8 = 10 times 8 = 80[/latex] cm² Answer: 80 cm² Example 9: Right Triangle Area Find the area of a right triangle with legs 6 cm and 8 cm. In a right triangle, the legs can be used as base and height Area = [latex]frac{1}{2} times 6 times 8 = 24[/latex] cm² Answer: 24 cm² Exercises (Triangles) Find the perimeter of a triangle with sides 7 cm, 9 cm, and 12 cm Calculate the area of a triangle with base 15 m and height 10 m A triangle has area 54 cm² and base 12 cm. Find its height Find the area of a right triangle with legs 5 cm and 12 cm A triangle has perimeter 30 cm and two sides are 8 cm and 11 cm. Find the third side Parallelogram - Area and Perimeter A parallelogram has opposite sides parallel and equal in length. Opposite angles are also equal. Parallelogram Formulas Perimeter = [latex]2 times (text{base} + text{side})[/latex] or [latex]2(a + b)[/latex] Area = [latex]text{base} times text{height}[/latex] or [latex]b times h[/latex] Important Note About Height ! The height of a parallelogram is the perpendicular distance between the base and the opposite side ! The height is NOT the same as the slanted side length ! Always use the perpendicular height when calculating area Example 10: Finding Parallelogram Perimeter Find the perimeter of a parallelogram with base 15 cm and side 8 cm. Perimeter = [latex]2 times (15 + 8) = 2 times 23 = 46[/latex] cm Answer: 46 cm Example 11: Finding Parallelogram Area Find the area of a parallelogram with base 18 m and height 7 m. Area = [latex]18 times 7 = 126[/latex] m² Answer: 126 m² Example 12: Finding Height from Area A parallelogram has area 96 cm² and base 12 cm. Find its height. Area = base × height [latex]96 = 12 times h[/latex] [latex]h = 96 div 12 = 8[/latex] cm Answer: Height = 8 cm Exercises (Parallelograms) Find the perimeter of a parallelogram with base 20 cm and side 13 cm Calculate the area of a parallelogram with base 25 m and height 6 m A parallelogram has area 144 cm² and height 9 cm. Find its base A parallelogram has perimeter 52 cm and base 16 cm. Find its side length Find the area of a parallelogram with base 14 cm and height 5.5 cm Shape Comparison and Relationships Shape Perimeter Formula Area Formula Key Feature Square [latex]4s[/latex] [latex]s^2[/latex] All sides equal Rectangle [latex]2(l + w)[/latex] [latex]l times w[/latex] Opposite sides equal Triangle [latex]a + b + c[/latex] [latex]frac{1}{2}bh[/latex] Three sides Parallelogram [latex]2(a + b)[/latex] [latex]b times h[/latex] Opposite sides parallel Important Relationships Square vs Rectangle: A square is a special rectangle where length = width Rectangle vs Parallelogram: A rectangle is a special parallelogram with all angles 90° Triangle vs Parallelogram: A triangle is half of a parallelogram with the same base and height Comparison Questions How is a square different from a rectangle? Why is the area of a triangle half the area of a parallelogram with the same base and height? What special type of parallelogram has all angles equal to 90°? Real-World Applications Understanding area and perimeter is essential in many everyday situations: Construction - Calculating materials for floors, walls, and fences Gardening - Planning garden beds and calculating soil needed Home Improvement - Buying paint, wallpaper, or flooring Sports - Marking out playing fields and courts Farming - Calculating land area for crops Example 13: Fencing a Garden A rectangular garden measures 12 m by 8 m. How much fencing is needed to enclose it? Perimeter = [latex]2 times (12 + 8) = 2 times 20 = 40[/latex] m Answer: 40 meters of fencing Example 14: Flooring a Room A square room has side length 5 m. How many square meters of flooring are needed? Area = [latex]5 times 5 = 25[/latex] m² Answer: 25 m² of flooring Example 15: Triangular Banner A triangular banner has base 3 m and height 2 m. What is its area? Area = [latex]frac{1}{2} times 3 times 2 = 3[/latex] m² Answer: 3 m² Real-World Problems A rectangular pool is 15 m long and 8 m wide. What is the perimeter of the pool? A square photo frame has area 144 cm². What is the length of each side? A triangular garden has base 10 m and height 6 m. How much soil is needed to cover it if 1 bag covers 2 m²? A parallelogram-shaped field has base 80 m and height 50 m. What is its area in hectares? (1 hectare = 10,000 m²) A rectangular room is 6 m by 4 m. How much carpet is needed to cover the floor? Cumulative Exercises Find the perimeter of a square with side 18 cm Calculate the area of a rectangle with length 14 m and width 9 m A triangle has sides 11 cm, 13 cm, and 16 cm. Find its perimeter Find the area of a parallelogram with base 22 cm and height 7 cm A square has perimeter 60 cm. What is its area? A rectangle has area 180 cm² and length 15 cm. Find its width A triangle has area 42 cm² and base 12 cm. Find its height A parallelogram has perimeter 48 cm and base 14 cm. Find its side length Find the area of a right triangle with legs 9 cm and 12 cm A rectangular garden is 12 m long and has perimeter 40 m. Find its width and area Which has greater area: a square of side 10 cm or a rectangle measuring 12 cm × 8 cm? A triangular flag has base 1.5 m and height 0.8 m. Find its area A parallelogram has area 135 m² and height 9 m. Find its base Find the perimeter of a triangle with sides in ratio 3:4:5 and longest side 20 cm A square and a rectangle have the same perimeter. The square has side 15 cm. The rectangle has length 20 cm. Find the area of the rectangle Show/Hide Answers Problem: Find the perimeter of a square with side 18 cm Step 1: Perimeter = [latex]4 times text{side}[/latex] Step 2: = [latex]4 times 18 = 72[/latex] cm Answer: 72 cm Problem: Calculate the area of a rectangle with length 14 m and width 9 m Step 1: Area = length × width Step 2: = [latex]14 times 9 = 126[/latex] m² Answer: 126 m² Problem: A triangle has sides 11 cm, 13 cm, and 16 cm. Find its perimeter Step 1: Perimeter = sum of all sides Step 2: = [latex]11 + 13 + 16 = 40[/latex] cm Answer: 40 cm Problem: Find the area of a parallelogram with base 22 cm and height 7 cm Step 1: Area = base × height Step 2: = [latex]22 times 7 = 154[/latex] cm² Answer: 154 cm² Problem: A square has perimeter 60 cm. What is its area? Step 1: Perimeter = [latex]4s = 60[/latex], so [latex]s = 60 div 4 = 15[/latex] cm Step 2: Area = [latex]s^2 = 15^2 = 225[/latex] cm² Answer: 225 cm² Problem: A rectangle has area 180 cm² and length 15 cm. Find its width Step 1: Area = length × width Step 2: [latex]180 = 15 times w[/latex] Step 3: [latex]w = 180 div 15 = 12[/latex] cm Answer: Width = 12 cm Problem: A triangle has area 42 cm² and base 12 cm. Find its height Step 1: Area = [latex]frac{1}{2} times text{base} times text{height}[/latex] Step 2: [latex]42 = frac{1}{2} times 12 times h[/latex] Step 3: [latex]42 = 6 times h[/latex] Step 4: [latex]h = 42 div 6 = 7[/latex] cm Answer: Height = 7 cm Problem: A parallelogram has perimeter 48 cm and base 14 cm. Find its side length Step 1: Perimeter = [latex]2 times (text{base} + text{side})[/latex] Step 2: [latex]48 = 2 times (14 + s)[/latex] Step 3: [latex]48 = 28 + 2s[/latex] Step 4: [latex]2s = 20[/latex], so [latex]s = 10[/latex] cm Answer: Side length = 10 cm Problem: Find the area of a right triangle with legs 9 cm and 12 cm Step 1: In a right triangle, legs can be base and height Step 2: Area = [latex]frac{1}{2} times 9 times 12 = 54[/latex] cm² Answer: 54 cm² Problem: A rectangular garden is 12 m long and has perimeter 40 m. Find its width and area Step 1: Perimeter = [latex]2 times (text{length} + text{width})[/latex] Step 2: [latex]40 = 2 times (12 + w)[/latex] Step 3: [latex]40 = 24 + 2w[/latex] Step 4: [latex]2w = 16[/latex], so [latex]w = 8[/latex] m Step 5: Area = [latex]12 times 8 = 96[/latex] m² Answer: Width = 8 m, Area = 96 m² Problem: Which has greater area: a square of side 10 cm or a rectangle measuring 12 cm × 8 cm? Step 1: Square area = [latex]10^2 = 100[/latex] cm² Step 2: Rectangle area = [latex]12 times 8 = 96[/latex] cm² Step 3: 100 > 96, so the square has greater area Answer: The square has greater area Problem: A triangular flag has base 1.5 m and height 0.8 m. Find its area Step 1: Area = [latex]frac{1}{2} times text{base} times text{height}[/latex] Step 2: = [latex]frac{1}{2} times 1.5 times 0.8 = 0.6[/latex] m² Answer: 0.6 m² Problem: A parallelogram has area 135 m² and height 9 m. Find its base Step 1: Area = base × height Step 2: [latex]135 = b times 9[/latex] Step 3: [latex]b = 135 div 9 = 15[/latex] m Answer: Base = 15 m Problem: Find the perimeter of a triangle with sides in ratio 3:4:5 and longest side 20 cm Step 1: Longest side corresponds to ratio 5 Step 2: Ratio factor = [latex]20 div 5 = 4[/latex] Step 3: Other sides = [latex]3 times 4 = 12[/latex] cm, [latex]4 times 4 = 16[/latex] cm Step 4: Perimeter = [latex]12 + 16 + 20 = 48[/latex] cm Answer: 48 cm Problem: A square and a rectangle have the same perimeter. The square has side 15 cm. The rectangle has length 20 cm. Find the area of the rectangle Step 1: Square perimeter = [latex]4 times 15 = 60[/latex] cm Step 2: Rectangle perimeter = [latex]2 times (20 + w) = 60[/latex] Step 3: [latex]20 + w = 30[/latex], so [latex]w = 10[/latex] cm Step 4: Rectangle area = [latex]20 times 10 = 200[/latex] cm² Answer: 200 cm² Conclusion/Recap In this lesson, we've explored area and perimeter of four important 2D shapes. Remember these key points: Perimeter is the distance around a shape (linear units) Area is the space inside a shape (square units) Square: Perimeter = [latex]4s[/latex], Area = [latex]s^2[/latex] Rectangle: Perimeter = [latex]2(l + w)[/latex], Area = [latex]l times w[/latex] Triangle: Perimeter = sum of sides, Area = [latex]frac{1}{2}bh[/latex] Parallelogram: Perimeter = [latex]2(a + b)[/latex], Area = [latex]b times h[/latex] These concepts are essential for many real-world applications from construction to gardening. 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