Area of Polygons | FIO

Question 20

Observe the parallelograms in the figure below.

(i) What can we say about the areas of all these parallelograms?

(ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?

Solution

We will calculate the area and perimeter for each parallelogram by using the grid to find their dimensions.

Step 1 — Identify Base and Height

A parallelogram is a four-sided shape with opposite sides parallel. The base is the length of one side. The height is the perpendicular distance between the base and its opposite side.

Let us look at each parallelogram (a) to (g). We can count the grid units for the base. The bottom side of each parallelogram is horizontal. Its length is 5 units. This is the base. The vertical distance between the top and bottom sides is 3 units. This is the height.

$\text{Base} = 5 \text{ units}$

$\text{Height} = 3 \text{ units}$

Diagram 1

Step 2 — Calculate the Area

The formula for the area of a parallelogram is base multiplied by height.

$\text{Area} = \text{Base} \times \text{Height}$

$\text{Area} = 5 \times 3$

$\boxed{15 \text{ sq. units}}$

Since all parallelograms have the same base (5 units) and the same height (3 units), their areas are all equal.

Step 3 — Determine Side Lengths for Perimeter

The perimeter of a parallelogram is the total length of all its four sides. Since opposite sides are equal, the perimeter is twice the sum of the lengths of two adjacent sides.

$\text{Perimeter} = 2 \times (\text{Base} + \text{Slanted Side Length})$

The base length is 5 units for all figures. We need to find the length of the slanted side. We can use the Pythagorean theorem for this. Imagine a right-angled triangle formed by the height, the horizontal shift, and the slanted side. Let 'h' be the height (3 units). Let 'x' be the horizontal shift of the top base relative to the bottom base. The length of the slanted side 's' is $\sqrt{x^2 + h^2}$ .

Let's find 'x' for each parallelogram:

(a): The top-left corner is 1 unit left of the bottom-left corner. So, $x = 1$ .
(b): The top-left corner is 2 units left of the bottom-left corner. So, $x = 2$ .
(c): The top-left corner is 3 units left of the bottom-left corner. So, $x = 3$ .
(d): This is a rectangle. The top-left corner is directly above the bottom-left corner. So, $x = 0$ .
(e): The top-left corner is 4 units left of the bottom-left corner. So, $x = 4$ .
(f): The top-left corner is 5 units left of the bottom-left corner. So, $x = 5$ .
(g): The top-left corner is 6 units left of the bottom-left corner. So, $x = 6$ .

Step 4 — Calculate and Compare Perimeters

Now we calculate the slanted side length and then the perimeter for each figure.

For figure (d) (rectangle): $\text{Slanted side length} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3 \text{ units}$

$\text{Perimeter (d)} = 2 \times (5 + 3) = 2 \times 8 = 16 \text{ units}$

For figure (a): $\text{Slanted side length} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \text{ units}$

$\text{Perimeter (a)} = 2 \times (5 + \sqrt{10}) \text{ units}$

For figure (b): $\text{Slanted side length} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \text{ units}$

$\text{Perimeter (b)} = 2 \times (5 + \sqrt{13}) \text{ units}$

For figure (c): $\text{Slanted side length} = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \text{ units}$

$\text{Perimeter (c)} = 2 \times (5 + \sqrt{18}) \text{ units}$

For figure (e): $\text{Slanted side length} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ units}$

$\text{Perimeter (e)} = 2 \times (5 + 5) = 2 \times 10 = 20 \text{ units}$

For figure (f): $\text{Slanted side length} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \text{ units}$

$\text{Perimeter (f)} = 2 \times (5 + \sqrt{34}) \text{ units}$

For figure (g): $\text{Slanted side length} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \text{ units}$

$\text{Perimeter (g)} = 2 \times (5 + \sqrt{45}) \text{ units}$

To compare the perimeters, we compare the lengths of the slanted sides. As the horizontal shift 'x' increases, the slanted side length $\sqrt{x^2 + 3^2}$ increases. This makes the perimeter larger. The smallest 'x' is 0 for figure (d). So, figure (d) has the minimum perimeter. The largest 'x' is 6 for figure (g). So, figure (g) has the maximum perimeter.

Answer

(i) All the parallelograms (a) to (g) have the same base (5 units) and the same height (3 units). The area of a parallelogram is base × height. So, Area = 5 × 3 = 15 sq. units. Therefore, all these parallelograms have equal areas. (ii) Their perimeters are different. As the slant of the sides increases (meaning the horizontal shift 'x' increases), the length of the slanted side increases, and thus the perimeter increases. Figure (d) (which is a rectangle, with no slant) has the minimum perimeter. Figure (g) (which has the most slanted sides) has the maximum perimeter.

More questions in FIO

Identify the missing sidelengths.

The figure shows a path (the shaded portion) laid around a rectangular park EFGH.

(i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign possible values of your choice to these measurements and find the area of the path. Give a formula for the area.

An example of a formula — $Area\ of\ a\ rectangle = length \times width$ .

[Hint: There is a relation between the areas of EFGH, the path, and ABCD.**]

(ii) If the width of the path along each side is given, can you find its area? If not, what other measurements do you need? Assign values of your choice to these measurements and find the area of the path. Give a formula for the area using these measurements.

[Hint: Break the path into rectangles.**]

(iii) Does the area of the path change when the outer rectangle is moved while keeping the inner rectangular park EFGH inside it, as shown?

The figure shows a plot with sides 14m and 12m, and with a crosspath. What other measurements do you need to find the area of the crosspath? Once you identify the lengths to be measured, assign some possible values of your choice and find the area of the path. Give a formula for the area based on the measurements you choose.

Find the area of the spiral tube shown in the figure. The tube has the same width throughout.

[Hint: There are different ways of finding the area. Here is one method.]

What should be the length of the straight tube if it is to have the same area as the bent tube on the left?

In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons.

Divide a square into 4 parts by drawing two perpendicular lines inside the square as shown in the figure.

Rearrange the pieces to get a larger square, with a hole inside.

You can try this activity by constructing the square using cardboard, thick chart paper, or similar materials.

Find the areas of the following triangles:

Find the length of the altitude BY.

Find the area of $\Delta$ SUB, given that it is isosceles, SE is perpendicular to UB, and the area of $\Delta$ SEB is 24 sq. units.

Q10

[Śulba-Sūtras] Give a method to transform a rectangle into a triangle of equal area.

Q11

[Śulba-Sūtras] Give a method to transform a triangle into a rectangle of equal area.

Q12

ABCD, BCEF, and BFGH are identical squares.

(i) If the area of the red region is 49 sq. units, then what is the area of the blue region?

(ii) In another version of this figure, if the total area enclosed by the blue and red regions is 180 sq. units, then what is the area of each square?

Q13

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Q14

Gopal needs to carry water from the river to his water tank. He starts from his house. What is the shortest path he can take from his house to the river and then to the water tank? Roughly recreate the map in your notebook and trace the shortest path.

Q15

Find the area of the quadrilateral ABCD given that AC = 22 cm, BM = 3 cm, DN = 3 cm, BM is perpendicular to AC, and DN is perpendicular to AC.

Q16

Find the area of the shaded region given that ABCD is a rectangle.

Q17

What measurements would you need to find the area of a regular hexagon?

Q18

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Q19

Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral.

Q20

Observe the parallelograms in the figure below.

(i) What can we say about the areas of all these parallelograms?

(ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?

Q21

Find the areas of the following parallelograms:

Q22

Find QN.

Q23

Consider a rectangle and a parallelogram of the same sidelengths: 5 cm and 4 cm. Which has the greater area? [Hint: Imagine constructing them on the same base.]

Q24

Give a method to obtain a rectangle whose area is twice that of a given triangle. What are the different methods that you can think of?

Q25

[Śulba-Sūtras] Give a method to obtain a rectangle of the same area as a given triangle.

Q26

[Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it?

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Q27

[Śulba-Sūtras] Give a method to convert a rectangle into an isosceles triangle by dissection.

Q28

Which has greater area—an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area—two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.

Q29

Find the area of a rhombus whose diagonals are 20 cm and 15 cm.

Q30

Give a method to convert a rectangle into a rhombus of equal area using dissection.

Q31

Find the areas of the following figures:

Q32

[Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.

Q33

Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area —

Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH?

[Hint: If ΔAHI ≅ ΔDGI and ΔBEJ ≅ ΔCFJ, then the trapezium and rectangle have equal areas.]

Q34

Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area $144\text{ cm}^2$ .

Q35

A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.

Q36

ZYXW is a trapezium with ZY || WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of $\Delta\text{ZWB}$ .