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California Standards

California Standards.

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California Standards

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  1. California Standards MG2.1 Use formulas routinely for finding the perimeter 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. Also covered:MG2.2, MG2.4, andMG3.2

  2. w l Perimeter is the distance around a polygon. To find the perimeter of any polygon, you add the lengths of all its sides. Since opposite sides of a parallelogram are equal in length, you can find a formula for the perimeter of a parallelogram. P = w + l+ w + l = w + w + l+ l = 2w + 2l

  3. 5 14 Additional Example 1: Finding the Perimeter of Parallelograms A. Find the perimeter of the figure. P = 2w + 2l Perimeter of a parallelogram. = 2(5) + 2(14) Substitute 5 for w and 14 for l. = 10 + 28 = 38 units

  4. 16 20 Additional Example 1: Finding the Perimeter of Parallelograms B. Find the perimeter of the figure. P = 2w + 2l Perimeter of a parallelogram. = 2(16) + 2(20) Substitute 16 for w and 20 for l. = 32 + 40 = 72 units

  5. Check It Out! Example 1 A. Find the perimeter of the figure. 6 11 P = 2w + 2l Perimeter of a parallelogram. = 2(6) + 2(11) Substitute 6 for w and 11 for l. = 12 + 22 = 34 units

  6. Check It Out! Example 1 B. Find the perimeter of the figure. 5 13 P = 2w + 2l Perimeter of a parallelogram. = 2(5) + 2(13) Substitute 5 for w and 13 for l. = 10 + 26 = 36 units

  7. Height Side Base The area of a plane figure is the number of unit squares needed to cover the figure. The baseof a parallelogram is the length of one side. The heightis the perpendicular distance from the base to the opposite side.

  8. While perimeter is expressed in linear units, such as inches (in.) or meters (m), area is expressed in square units, such as square feet (ft2). You can cut a parallelogram and shift the cut piece to form a rectangle whose base and height are the same as those of the original parallelogram. The same number of unit squares are needed to cover the two figures. So a parallelogram and a rectangle that have the same base and height have the same area.

  9. Helpful Hint Since the base and height of a rectangle are the same as its length and width, the formula for the area of a rectangle can also be written as A = lw.

  10. Additional Example 2: Using a Graph to Find Area Graph and find the area of the figure with the given vertices. A. (–1, –2), (2, –2), (2, 3), (–1, 3) Area of a rectangle. A = bh Substitute 3 for b and 5 for h. A = 3 • 5 A = 15 units2

  11. Caution! The height of a parallelogram is not the length of its slanted side. The height of a figure is always perpendicular to the base.

  12. Additional Example 2: Using a Graph to Find Area Graph and find the area of the figure with the given vertices. B. (0, 0), (5, 0), (6, 4), (1, 4) Area of a parallelogram. A = bh Substitute 5 for b and 4 for h. A = 5 • 4 A = 20 units2

  13. y (–3, 3) (1, 3) x 5 4 (1, –2) (–3, –2) Check It Out! Example 2 Graph and find the area of the figure with the given vertices. A. (–3, –2), (1, –2), (1, 3), (–3, 3) Area of a rectangle. A = bh Substitute 4 for b and 5 for h. A = 4 • 5 A = 20 units2

  14. y (1, 3) (5, 3) x 4 (3, –1) 4 (–1, –1) Check It Out! Example 2 Graph the figure with the given vertices. Then find the area of the figure. B. (–1, –1), (3, –1), (5, 3), (1, 3) Area of a parallelogram. A = bh Substitute 4 for b and 4 for h. A = 4 • 4 A = 16 units2

  15. A composite figureis made up of basic geometric shapes such as rectangles, triangles, trapezoids, and circles. To find the area of a composite figure, find the areas of the geometric shapes and then add the areas.

  16. Additional Example 3: Finding Area and Perimeter of a Composite Figure Find the perimeter and area of the figure. 6 6 3 3 6 5 5 The length of the side that is not labeled is the same as the sum of the lengths of the sides opposite, 18 units. P = 5 + 6 + 3 + 6 + 3 + 6 + 5 + 18 = 52 units

  17. Additional Example 3 Continued 6 6 3 3 6 5 5 A = 6 • 5 + 6 • 2 + 6 • 5 Add the areas together. = 30 + 12 + 30 = 72 units2

  18. Check It Out! Example 3 Find the perimeter of the figure. The length of the side that is not labeled is 2. 2 4 6 7 7 2 6 2 P = 6 + 2 + 4 + 7 + 6 + 4 + 2 + 2 + 2 + 7 ? = 42 units 4

  19. 2 4 7 2 6 2 2 2 Check It Out! Example 3 Continued 2 Find the area of the figure. 4 6 7 Add the areas together. A = 2 • 6 + 7 • 2 + 2 • 2 + 4 • 2 7 2 6 2 = 12 + 14 + 4 + 8 2 2 = 38 units2 4 + + +

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