Section 8.5

Find the area of parallelograms.

Base of a parallelogram Height of a parallelogram Parallelogram Rhombus

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).

Either pair of parallel sides of a parallelogram are called the bases of the parallelogram. base

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.

What is the area of this parallelogram? b=base h=height

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

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

It’s the same as the area of this rectangle h=height b=base

Area rectangle = base x height (perpendicular height h) h=height b=base

Area parallelogram = base x height (perpendicular height h) b=base h=height

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

Find the Area of a Parallelogram Example 1 Find the area of the parallelogram. SOLUTION Use the formula for the area of a parallelogram. Substitute 9 for b and 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

Find the Height of a Parallelogram Example 2 Find the height of the parallelogram given that its area is 78 square feet. SOLUTION 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

Your Turn: Find the area of the parallelogram ANSWER 96 yd 2 ANSWER 77 mm 2 ANSWER 196 ft 2

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

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

Same as Area of parallelogram A = b. h b h

Since the diagonals are perpendicular Another way to find the area of a rhombus is: Area = ½ (product of the diagonals) d1d1 d2d2 area = ½ (d 1. d 2 ) This is good when you only know the diagonals, but not the sides or height

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 1/2d 1 d 2 or ½(6)(8), also equals 24 The formula for the area of a rhombus can be justified using the area of a triangle. A specific case follows.

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

Pg. 442 – 445: #1 – 27 odd, 33 – 47 odd