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Measuring Quadrilaterals

Objective

In the upcoming section, we’ll review a variety of formulas to measure the perimeter and area of quadrilaterals.

Previously Covered

  • A quadrilateral is a four-sided polygon.
  • A parallelogram is a quadrilateral whose opposite sides are parallel.
  • A rectangle is a parallelogram with four right angles.
  • A rhombus is a parallelogram whose sides are all congruent.
  • A square is both a rhombus and a rectangle.
  • A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Quadrilaterals: Perimeter

Recall that the perimeter of a polygon is the sum of the lengths of its sides. Questions involving the perimeter of a parallelogram may rely on your knowledge of the properties of that parallelogram. Let’s look at some examples.

Question

Which of the following gives the perimeter of a rhombus with side length s?

  1. P = s2
  2. P = 2s2
  3. P = 4s
  4. P = 4s2

Reveal Answer

Choice C is the correct answer. The perimeter of a polygon is the sum of the lengths of its sides. Since all four sides of a rhombus are congruent, the perimeter is s + s + s + s, or 4s. In fact, you could have eliminated all of the other choices, since they give measures in square units.

Question

Which of the following gives the perimeter of a rectangle with length l and width w?

  1. P = 2lw
  2. P = 4(lw)
  3. P = 2(l + w)
  4. P = 4(l + w)

Reveal Answer

Choice C is the correct answer. Since the opposite sides of a rectangle are congruent, they have the same measure. So the perimeter of a rectangle is equal to the sum of 2 lengths (2l) and 2 widths (2w): 2l + 2w = 2(l + w).

Quadrilaterals: Area

Parallelograms: The area of a parallelogram is given by the formula A = bh, where b is the base and h is the height. Again, be careful. Don’t forget that the base and the height must be perpendicular.

Parallelogram
Looking Back: Do you now see why the area of a triangle is A= 1/2bh?

Question

What is the area of this parallelogram?

Parallelogram area question

  1. 4 cm2
  2. 8 cm2
  3. 9 cm2
  4. 14 cm2

Reveal Answer

Choice B is correct. The area of a parallelogram is , where b is the base and h is the height. The base of this parallelogram is 4 cm long. Because the height must be perpendicular to the base, the height is 2 cm. If you answered A, you may have used the formula for the area of a triangle rather than the area of a parallelogram. If you answered D, you may have been finding the perimeter rather than the area. So the area is:

Answer equation

Trapezoids

The area of a trapezoid is given by the formula Formula for area of trapezoid, where h is the height and b1 and b2 are the lengths of the parallel bases.

Area of a trapezoid

Question

What is the area of the trapezoid below? Round your answer to the nearest tenth of a square centimeter.

Trapezoid area question

  1. 12 cm2
  2. 13.5 cm2
  3. 27 cm2
  4. 30.5 cm2

Reveal Answer

Choice B is the correct answer. The area of the trapezoid is given by the formula Formula for area of trapezoid, where h = 3 cm, b1 = 4 cm, and b2 = 5 cm. So the area of the trapezoid is:
Answer equation

Review

  • The perimeter of a quadrilateral is the sum of the lengths of its sides.
  • The area of a parallelogram is given by the formula, A= bh where b is the base and h is the height and the base and height are perpendicular.
  • The area of a trapezoid is given by the formula Formula for area of trapezoid, where h is the height and b1 and b2 are the lengths of the parallel bases.

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