Area of Parallelograms

1. Area of Parallelograms Section 11.2

2. Goal • Find the area of parallelograms.

3. Key Vocabulary • Base of a parallelogram • Height of a parallelogram • Parallelogram • Rhombus

4. Parallelogram • A parallelogram is a quadrilateral where the opposite sides are congruent and parallel. • A rectangle is a type of parallelogram, but we often see parallelograms that are not rectangles (parallelograms without right angles).

5. The Base of a Parallelogram • Either pair of parallel sides of a parallelogram are called the bases of the parallelogram. base base base base

6. The Height of a Parallelogram • The shortest distance (perpendicular distance) between the bases of a parallelogram is called the height of the parallelogram. • The height of the parallelogram is always perpendicular to the bases.

7. Area of Parallelogram What is the area of this parallelogram? h=height b=base

8. Area of Parallelogram What is the area of this parallelogram? h=height b=base CUT HERE!

9. Area of Parallelogram What is the area of this parallelogram? h=height b=base MOVE TO HERE!

10. Area of Parallelogram It’s the same as the area of this rectangle h=height b=base

11. Area of Parallelogram Arearectangle= base x height (perpendicular height h) h=height b=base

12. Area of Parallelogram Areaparallelogram = base x height (perpendicular height h) h=height b=base

13. Area of Parallelogram • Words: Area =(base)(height) • Symbols: A = bh h b Very Important: The height must be perpendicular to the base.

14. SOLUTION Use the formula for the area of a parallelogram. Substitute 9 for band 6 for h. A = bh Formula for the area of a parallelogram =(9)(6) Substitute 9 for b and 6 for h. =54 Multiply. The parallelogram has an area of 54 square meters. ANSWER Example 1 Find the Area of a Parallelogram Find the area of the parallelogram.

15. A =bh Formula for the area of a parallelogram 78=12h Substitute 78 for A and 12 for b. 6.5 =h Divide each side by 12. The parallelogram has a height of 6.5 feet. ANSWER Example 2 Find the Height of a Parallelogram Find the height of the parallelogram given that its area is 78 square feet. SOLUTION

16. Your Turn: 1. 96 yd2 ANSWER 2. 77 mm2 ANSWER 3. 196 ft2 ANSWER Find the area of the parallelogram.

17. Your Turn: 4. A = 72 in.2 h =6 in. ANSWER A = 30 m2 5. b =6 m ANSWER 6. A = 28 cm2 h =4 cm ANSWER In Exercises 4–6, Agives the area of the parallelogram. Find the missing measure.

18. Rhombus • A parallelogram with opposite equal acute and obtuse angles and four equal sides. Diagonals 4 equal sides Diagonals

19. h b Area of a Rhombus Same as Area of parallelogram A = b . h

20. d2 d1 Special “Rhombus Rule” • Since the diagonals are perpendicular • Another way to find the area of a rhombus is: • Area = ½ (product of the diagonals) area = ½ (d1.d2) This is good when you only know the diagonals, but not the sides or height

21. Area of a Rhombus The formula for the area of a rhombus can be justified using the area of a triangle. A specific case follows. • The diagonals divide a rhombus into 4 congruent right triangles. So, the area of the rhombus is 4 times the area of one of the right triangles. • Area of 1 triangle = 1/2bh = ½(3)(4) = 6 • Area of 4 triangles = 4(6) = 24 • Notice that 1/2d1d2 or ½(6)(8), also equals 24

22. Find the area of the rhombus. b. a. 1 1 1 1 1 2 2 2 2 2 Example 3 Find the Area of a Rhombus SOLUTION SOLUTION a. b. A=d1d2 A=d1d2 = (6 + 6)(9 + 9) = (14)(10) = 70 = (12)(18) =108 The area of the rhombus is 70 square inches. The area of the rhombus is 108 square inches.