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Lesson 3.1.2

Areas of Polygons. Lesson 3.1.2. Lesson 3.1.2. Areas of Polygons. California Standard: Measurement and Geometry 1.2

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Lesson 3.1.2

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  1. Areas of Polygons Lesson 3.1.2

  2. Lesson 3.1.2 Areas of Polygons California Standard: Measurement and Geometry 1.2 Use formulas routinely for finding theperimeter and area of basic two-dimensional figures, and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. What it means for you: You’ll use formulas to find the areas of regular shapes. • Key words: • area • triangle • parallelogram • trapezoid • formula • substitution

  3. Perimeter Area Lesson 3.1.2 Areas of Polygons Area is the amount of space inside a shape. Like for perimeter, there are formulas for working out the areas of some polygons. You’ll practice using some of them in this Lesson.

  4. Parallelogram: Rectangle: A = bh A = lw Square: s w h A = s2 s l b Lesson 3.1.2 Areas of Polygons Area is the Amount of Space Inside a Shape Areais the amount of surface covered by a shape. Parallelograms, rectangles, and squares all have useful formulas for finding their areas. Triangles and other shapes can be a little more difficult, but there are formulas for those too — which we’ll come to next.

  5. 2 in 7 in Lesson 3.1.2 Areas of Polygons Example 1 Use a formula to evaluate the area of this shape. Solution Use the formula for the area of a rectangle. A = lw Substitute in the values given in the question. A = 7 in × 2 in Evaluate the area. A = 14 in2 Solution follows…

  6. Example 2 bh h b Lesson 3.1.2 Areas of Polygons You can also rearrange the formulas to find a missing length: Find the height of a parallelogram of area 42 cm2and base length 7 cm. Solution Rearrange the formula for the area of a parallelogram, and substitute. A = bh A ÷ b = bh÷ b = h Divide both sides by the base (b) h = 42 ÷ 7 = 6 cm Substitute values and evaluate Solution follows…

  7. Lesson 3.1.2 Areas of Polygons Guided Practice 1. Find the area of a square of side 2.4 m. 2. Find the length of a rectangle if it has area 30 in2, and width 5 in. A = s2 = 2.42 = 5.76 m2 A = lwl = A ÷ w = 30 ÷ 5 = 6 in Solution follows…

  8. height (h) + height (h) base (b) base (b) Lesson 3.1.2 Areas of Polygons The Area of a Triangle is Half that of a Parallelogram The area of a triangle is half the area of a parallelogram that has the same base lengthandvertical height.

  9. Area of triangle = area of parallelogram = (base × height) = bh 1 1 1 1 A = bh 2 2 2 2 Lesson 3.1.2 Areas of Polygons In math language, the area of a triangle is given by:

  10. 8 in b A = bh 1 2 Lesson 3.1.2 Areas of Polygons Example 3 Find the base length of the triangle shown if it has an area of 20 in2 and a height of 8 in. Solution Now substitute in the values and evaluate to give the base length. Rearrange the formula for the area to give an expression for the base length of the triangle. b = 2A÷h Write out the formula = (2 × 20) ÷ 8 2A = bh Multiply both sides by 2 2A ÷ h= bh ÷ h Divide both sides by the height (h) = 5 in Simplify b = 2A ÷ h Solution continues… Solution follows…

  11. Lesson 3.1.2 Areas of Polygons Guided Practice 3. Find the area of a triangle of base length 3 ft and height 4.5 ft. 4. Find the base length of a triangle with height 50 m and area 400 m2. A = 0.5(3 × 4.5) = 0.5 • 13.5 = 6.75 ft2 b = 2A ÷ h = (2 • 400) ÷ 50 = 16 m Solution follows…

  12. base of triangle 2 (b2) height (h) 2 2 1 1 base of triangle 1 (b1) Lesson 3.1.2 Areas of Polygons Break a Trapezoid into Parts to Find its Area The most straightforward way to find the area of a trapezoid is to split it up into two triangles. You then have to work out the area of both triangles and add them to find thetotal area. Notice that both triangles have the same height but different bases.

  13. b2 h b1 Area of trapezoid = area of triangle 1 + area of triangle 2 = b1h+ b2h 2 Take out the common factor of h to give: 1 Area of trapezoid = h(b1 + b2) 1 1 1 1 1 A = h(b1 + b2) 2 2 2 2 2 Lesson 3.1.2 Areas of Polygons So, the area of the trapezoid is the sum of the areas of each triangle.

  14. 12 ft 8 ft 30 ft Area of trapezoid = h(b1 + b2) Area of trapezoid = × 8 × (12 + 30) = × 8 × 42 = 168 ft2. 1 1 1 2 2 2 Lesson 3.1.2 Areas of Polygons Example 4 Find the area of the trapezoid shown. Solution Substitute in the values given in the question and evaluate. Solution follows…

  15. 5 in 20 cm 3 in 11 cm 10 in 4 cm 1.1 m 105 ft 80 ft 0.7 m 1.5 m 245 ft Lesson 3.1.2 Areas of Polygons Guided Practice Find the areas of the trapezoids in Exercises 5–8, using the formula.5. 6. 7. 8. 0.5 • 3 • (5 + 10)=22.5 in2 0.5 • 11 • (20 + 4)= 132 cm2 0.5 • 0.7 • (1.1 + 1.5)= 0.91 m2 0.5 • 80 • (105 + 245)= 14,000 ft2 Solution follows…

  16. 1 m 1.2 ft 1 m 1.2 ft Lesson 3.1.2 Areas of Polygons Independent Practice Find the area of each of the shapes in Exercises 1–2. 1. 2. 1.44 ft2 0.5 m2 Solution follows…

  17. 2.5 in 7 in 2.3 in 2 in Lesson 3.1.2 Areas of Polygons Independent Practice Find the area of each of the shapes in Exercises 3–4. 3. 4. 2.3 in2 17.5 in2 Solution follows…

  18. 11 cm 3.1 ft 12 cm 4.5 ft 20 cm Lesson 3.1.2 Areas of Polygons Independent Practice Find the area of each of the shapes in Exercises 5–6. 5. 6. 186 cm2 13.95 ft2 Solution follows…

  19. 3.1 m 2.4 m Lesson 3.1.2 Areas of Polygons Independent Practice 7. Miguel wants to know the area of his flower bed, shown below. Find the area using the correct formula. 3.72 m2 Solution follows…

  20. Lesson 3.1.2 Areas of Polygons Round Up Later you’ll use these formulas to find the areas of irregular shapes. Make sure you practice all this stuff so that you’re on track for the next few Lessons.

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