Download App

Perimeter and Area — MATHS STD 6 — Question

Gujarat BoardEnglish MediumSTD 6MATHSPerimeter and Area5 Marks
Question
On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 12 square units.
(a) Which rectangle has the greatest perimeter?
(b) Which rectangle has the least perimeter?
(c) If you take a rectangle of area 32 sq cm, what will your answers be? Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answer.
✓

Answer


Image
Perimeter of (a) = 2(1 + 24) = 2 × 25 = 50 units
Perimeter of (b) = 2(2 + 12) = 2 × 14 = 28 units
Perimeter of (c) = 2(4 + 6) = 2 × 10 = 20 units
Perimeter of (d) = 2(3 + 8) = 2 × 11 = 22 units
(a) Clearly rectangle (a) has the greatest perimeter.
(b) Obviously rectangle (c) has the least perimeter.
(c) Yes, it is possible to predict the shape of a rectangle with the greatest and least perimeter for a given area. Here’s how:
Greatest Perimeter: For a given area, the rectangle with the greatest perimeter will have one side as small as possible. This essentially means that the rectangle becomes very elongated.
For example, if the area is 24 square units, a rectangle with dimensions 1 unit by 24 units will have the greatest perimeter.
Example: Area = 24 square units
Dimensions = 1 unit by 24 units
Perimeter = 2(1 + 24) = 50 units
Least Perimeter: The rectangle with the least perimeter for a given area will be as close to a square as possible. This is because a square has the smallest perimeter for a given area among all rectangles.
Example: Area = 24 square units
Dimensions = 4 units by 6 units (since 4 × 6 = 24)
Perimeter = 2(4 + 6) = 20 units
Reasoning
Greatest Perimeter: When one side is minimized, the other side must be maximized to maintain the same area. This increases the sum of the sides, thus increasing the perimeter.
Least Perimeter: A square or a shape close to a square minimizes the sum of the sides for a given area, thus minimizing the perimeter.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Perimeter and Area→Perimeter and Area — all question types→MATHS STD 6 question papers→Gujarat Board STD 6 question papers→Gujarat Board question paper generator→

Similar questions

Arrange the following numbers in ascending order:
$56943201, 5694437, 56944000, 5695440, 56943300$
→
Match the following:
$(a)$ Straight angle $(i)$ Less than one-fourth a revolution
$(b)$ Right angle $(ii)$ More than half a revolution
$(c)$ Acute angle $(iii)$ Half of a revolution
$(d)$ Obtuse angle $(iv)$ One-fourth a revolution
$(v)$Reflex angle $(v)$Between $\frac{1}{4}$ and $\frac{1}{2}$ of a revolution
  $(vi)$One complete revolution
→
Draw $\angle\text{POQ}$ of measure $75^\circ $ and find its line of symmetry.
→
Use number line and add the integers: $(-1) + (-7).$
→
Draw a line segment $PQ = 6.2\ cm$. Draw the perpendicular bisector of $PQ.$
→
Draw an angle of $140^\circ$ with the help of a protractor and bisect it using ruler and compasses.
→
Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other.
→
Draw a line $l$ and a point $X$ on it.Through $X,$ draw a line segment $\overline{X Y}$ perpendicular to $l.$ Now draw a perpendicular to $\overline{X Y}$ at $Y. ($use ruler and compasses$)$
→
Construct with ruler and compasses, angles of measure: $120^\circ $
→
Akshi and Toshi start running along the rectangular tracks as shown in the figure. Akshi runs along the outer track and completes 5 rounds. Toshi runs along the inner track and completes 7 rounds. Now, they are wondering who ran more. Find out who ran the longer distance.
Image
→