Measuring Space: Perimeter and Area | EOT

Question 25

Fig. 6.53 shows two circles passing through each other's centres. Find the area of the region enclosed by the two circles in terms of the common radius $r$ .

Solution

Understand the Question

Two circles with the same radius $r$ have centers $O_1$ and $O_2$ . Since each circle passes through the center of the other, the distance between centers is $O_1O_2 = r$ .
Let the two intersection points of the circles be $A$ and $B$ . Connecting the centers and the intersection points forms two equilateral triangles, $\triangle O_1AO_2$ and $\triangle O_1BO_2$ , each of side length $r$ .
The central angle subtended by the common chord $AB$ at each center is $\angle AO_1B = 60^\circ + 60^\circ = 120^\circ$ .
The region enclosed between the two circles (common overlapping region) is equal to twice the area of the circular segment formed by the chord $AB$ , or the sum of the areas of two sectors minus the area of the rhombus $O_1AO_2B$ .

Step 1 · Find the Central Angle Subtended by the Common Chord

Let the centers of the two circles be $O_1$ and $O_2$ , and their points of intersection be $A$ and $B$ .In $\triangle O_1AO_2$ : $O_1A = O_2A = O_1O_2 = r$

Thus, $\triangle O_1AO_2$ is an equilateral triangle, so: $\angle AO_1O_2 = 60^\circ$

Similarly, in $\triangle O_1BO_2$ : $O_1B = O_2B = O_1O_2 = r \implies \angle BO_1O_2 = 60^\circ$

Therefore, the total central angle subtended by chord $AB$ at center $O_1$ is: $\angle AO_1B = \angle AO_1O_2 + \angle BO_1O_2 = 60^\circ + 60^\circ = 120^\circ$

Step 2 · Find the Area of One Circular Segment

The area of the circular segment bounded by chord $AB$ and arc $AB$ in circle $O_1$ is: $\text{Area of segment} = \text{Area of sector } O_1AB - \text{Area of } \triangle O_1AB$

Calculate the area of sector $O_1AB$ :

\begin{aligned} \text{Area of sector } O_1AB &= \dfrac{\theta}{360^\circ} \times \pi r^2 \\[0.6em] &= \dfrac{120^\circ}{360^\circ} \times \pi r^2 \\[0.6em] &= \dfrac{1}{3}\pi r^2 \end{aligned}

Calculate the area of $\triangle O_1AB$ :

\begin{aligned} \text{Area of } \triangle O_1AB &= \dfrac{1}{2} r^2 \sin 120^\circ \\[0.6em] &= \dfrac{1}{2} r^2 \times \dfrac{\sqrt{3}}{2} \\[0.6em] &= \dfrac{\sqrt{3}}{4} r^2 \end{aligned}

Therefore, the area of one segment is: $\text{Area of segment} = \dfrac{1}{3}\pi r^2 - \dfrac{\sqrt{3}}{4}r^2$

Step 3 · Calculate the Total Enclosed Area

The region enclosed by both circles consists of two such identical segments (one on each side of chord $AB$ ):

\begin{aligned} \text{Total enclosed area} &= 2 \times \text{Area of one segment} \\[0.6em] &= 2 \left(\dfrac{1}{3}\pi r^2 - \dfrac{\sqrt{3}}{4}r^2\right) \\[0.6em] &= \left(\dfrac{2\pi}{3} - \dfrac{\sqrt{3}}{2}\right) r^2 \\[0.6em] &= \left(\dfrac{4\pi - 3\sqrt{3}}{6}\right) r^2 \end{aligned}

Answer

$\left(\dfrac{4\pi - 3\sqrt{3}}{6}\right) r^2$

Common Mistakes

Incorrect Central Angle: Using $\theta = 60^\circ$ instead of $120^\circ$ for sector $O_1AB$ by only considering $\triangle O_1AO_2$ rather than the full angle $\angle AO_1B$ .
Missing Triangle Area: Forgetting to subtract the triangle's area when finding the segment area, mistakenly using just the sector area.
Single Segment Only: Forgetting to multiply the single segment area by $2$ to account for both symmetric halves of the overlapping lens.

More questions in EOT

Identities in algebra can sometimes be shown as area relationships. For example:

The figure shown corresponds to the identity $(a + b)^2 = a^2 + 2ab + b^2$

Do you see how?

Draw figures corresponding to the identities $(a + b)(a - b) = a^2 - b^2$ and $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ .

An isosceles triangle has perimeter $40\text{ cm}$ ; the equal sides are $15\text{ cm}$ each. Find the area of the triangle.

An isosceles triangle has base $10\text{ cm}$ , and its area is $60\text{ cm}^2$ . What are the lengths of the equal sides?

The area of a right-angled triangle is $54\text{ sq. cm}$ . One of its legs has length $12\text{ cm}$ . Find its perimeter.

The sides of a triangle are in the ratio $2 : 3 : 4$ , and its perimeter is $45\text{ cm}$ . Find its area.

The sides of a triangle have lengths $7\text{ cm}$ , $24\text{ cm}$ , $25\text{ cm}$ . Find the area of the triangle in two different ways.

If the wheel of a bicycle has a diameter of $60\text{ cm}$ , find how far a cyclist will have travelled after the wheel has rotated $100$ times.

Find the area of a quadrant of a circle whose circumference is $66 \text{ cm}$ .

The wheel of a car has an outer radius of $28\text{ cm}$ . Calculate how far the car travels after one complete turn of the wheel, and how many times the wheel turns during a journey of $1\text{ km}$ .

Q10

Two rectangles have the same area and the same perimeter. Does this mean that they are congruent to each other?

Q11

You know that the area of a parallelogram is base $\times$ height. Using this and the figure, show that the area of a trapezium is half the sum of the parallel sides $\times$ height, i.e., $\dfrac{1}{2}(a + b)h$ .

Q12

By dividing a trapezium into two triangles show that its area is, half the sum of the parallel sides multiplied by the height (the same formula as the one given above).

Q13

Show how we can use two identical copies of a trapezium to make a parallelogram. How will this give us the formula for the area of a trapezium?

Q14

Show that the area of a kite is half the product of its diagonals. Show this: (i) using algebra, and (ii) using geometry.

Q15

Three problems about fitting congruent shapes together:

(i) Rectangle ABCD has sides $a$ , $b$ , and rectangle PQRS has sides $2a$ , $2b$ . Show that PQRS has 4 times the area of ABCD. Does this mean that 4 copies of rectangle ABCD will fit into rectangle PQRS? Check and see!

(ii) $\Delta ABC$ has sides $a$ , $b$ , $c$ , and $\Delta PQR$ has sides $2a$ , $2b$ , $2c$ . Show that $\Delta PQR$ has 4 times the area of $\Delta ABC$ . Does this mean that 4 copies of $\Delta ABC$ will fit into $\Delta PQR$ ? Check and see!

(iii) $\Delta ABC$ has sides $a$ , $b$ , $c$ , and $\Delta PQR$ has sides $3a$ , $3b$ , $3c$ . Show that $\Delta PQR$ has 9 times the area of $\Delta ABC$ . Does this mean that 9 copies of $\Delta ABC$ will fit into $\Delta PQR$ ? Check and see!

Q16

Find the fraction of the shaded region in each of the following figures:

(i) Fig. 6.43: What fraction of the triangle is shaded? (ii) Fig. 6.44: What fraction of the square is shaded?

Q17

Find the fraction of the rectangle covered by the circles in each of the following figures:

(i) Fig. 6.45: What fraction of the rectangle is covered by the circles?

(ii) Fig. 6.46: What fraction of the rectangle is covered by the circles?

Q18

Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture!

Q19

*19. The figure shows nine identical rectangles fitted together to make a large rectangle whose area is $72\text{ cm}^2$ . Find the perimeter of each small rectangle.

Q20

Show that the areas of the shaded blue triangle and the shaded red triangle are equal.

Find a way of cutting up the blue triangle into some number of pieces and rearranging the pieces to cover the red triangle.

Q21

The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two adjacent vertices. There are two semicircles on two adjacent sides as diameters. They create the shaded regions $A$ and $B$ .

Show that $A$ and $B$ have equal area.

Q22

In Fig. 6.50, four semicircles have been drawn within the given square whose side is $2 \text{ units}$ . The centres of these semicircles are the midpoints of the sides. They create a 4-petalled flower (shown in blue). Find the perimeter and the area of this flower.

Q23

In Fig. 6.51 we see two concentric circles with a common centre $O$ . A chord $BC$ of the larger circle is drawn, touching the smaller circle at $A$ . The length of $BC$ is $l$ . Show that the area of the green region enclosed between the two circles is $\dfrac{1}{4} \pi l^2$ .

Q24

In Fig. 6.52, semicircles have been drawn on all the sides of a right-angled triangle as shown. Show that $\text{Area (A)} + \text{Area (B)} = \text{Area (C)}$ .

Q25

Fig. 6.53 shows two circles passing through each other's centres. Find the area of the region enclosed by the two circles in terms of the common radius $r$ .

Q26

In Fig. 6.54, we see three triangles within a rectangle. The areas of the triangles are $A, B, C$ , as marked. Show that the area of the rectangle is

$\dfrac{2(A + C)(B + C)}{C}$

Q27

In the figure we see two shaded regions formed by a quarter circle, a semicircle, and a triangle.

Show that the areas of the two shaded regions are equal.