How to Find Areas Related to Circles and Squares?

By Team Aakash Byju's | 20th December 2022

From each corner of a square of side 4 cm, a quadrant of a circle of radius 1 cm is cut & also a circle of diameter 2 cm is cut as shown. Find the area of the remaining portion of the square.

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The area of the remaining portion of the given image is 9.72 square centimetres. Let us see the step-by-step calculation.

[{"selector":"#anim-18b20367-6c02-450d-8623-e5494dcc0880","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-15e4becb-529f-47ed-8b96-56915c7315ad","keyframes":{"transform":["translate3d(-115.55555%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}]

Given that the square given is of a side length of 4 centimetres.

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It is also given that all 4 corners of the squares represent quadrants of circles with a radius of 1 cm.

[{"selector":"#anim-1cd88237-6a60-4746-94cf-dc74cf76bf01","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-46319065-4e14-4419-aa51-805ebf6fe312","keyframes":{"transform":["translate3d(-115.55555%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-29744149-e866-42dc-9b3b-f69bf7c2cb3f","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-b59b1ae3-df1e-41fa-b05b-fac8a0011a14","keyframes":{"transform":["translate3d(-116.29393%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-59cb3285-cc7d-472f-824b-ab4e83f22a26","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-24a6da63-9ae7-407a-8820-bb886ec84e3b","keyframes":{"transform":["translate3d(-1094.99976%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-cd463ae6-8d5f-404a-a50d-c7acda11a609","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-ee153b39-5cdd-457d-aa67-97d42e1b136c","keyframes":{"transform":["translate3d(-1079.99988%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-c9e53bf6-46be-41ed-8709-c7eb1042d9dc","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-97d8284f-ed70-4f59-9d49-baa04cfe15c1","keyframes":{"transform":["translate3d(-1094.99976%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] π 2

So, the area of all 4 quadrants can be calculated using the area of sector formula:

[{"selector":"#anim-e06c887e-d731-4d2f-852c-114f9766dbc1","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-f8a14007-7af9-4d52-a763-ab939474354a","keyframes":{"transform":["translate3d(-115.55555%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] θ 360 × π × r 2

Given θ =

[{"selector":"#anim-f1b9a052-0268-45ef-9990-b579accbc4c3","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-f5105252-b9b6-425b-b7c1-c0a436dfd312","keyframes":{"transform":["translate3d(-157.30337%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-e8e3283c-f8a5-4f17-8ce4-71ac93448c7e","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-e89940eb-55f6-4b62-b3b8-ec5ec8f1b107","keyframes":{"transform":["translate3d(-829.99977%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-ec6cf7e2-0bba-4141-9f07-d2f6ddb40fcd","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-1f3fe6f8-2239-4be5-9bd1-f46821bc3eb5","keyframes":{"transform":["translate3d(-834.9998%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-58507f61-5208-42ce-9956-1be4c5a285d4","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-4b7acb76-9bae-4061-843d-a987ff15f398","keyframes":{"transform":["translate3d(-664.00009%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-ff9bf960-f7cc-4eec-abdd-9189fc66fe18","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-03f7038b-f20b-4fbf-8dd1-07628fcd30cd","keyframes":{"transform":["translate3d(-541.02558%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-06c17a73-b471-43f6-9f07-d59ba9846e01","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-086913f6-2e3f-4e2d-9738-8664b37c85cf","keyframes":{"transform":["translate3d(-285.08773%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-bc3e1334-4c89-4ec8-83a9-e2965e2c3b1b","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-707948bb-4d06-481d-959a-488d9cab953d","keyframes":{"transform":["translate3d(-167.10527%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-4de33f61-22fa-4423-b244-5d877ba6069c","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-1e3465b1-93c5-46da-8d74-3b78360a9568","keyframes":{"transform":["translate3d(-552.17429%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] 90 360 × π × π = 3.14cm 2 =90 4 × 1 2 , × 1

Also, there is a cut in the centre of the square which is a circle of radius 1 cm.

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So, the area of the remaining portion of the square is:

[{"selector":"#anim-0ff570b6-8728-433d-a854-ad0d900fd419","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-da075b5a-91ab-4d28-844b-0d842eea6b44","keyframes":{"transform":["translate3d(-115.55555%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-6a05903a-a810-4e3b-b9b0-4264e406bb21","keyframes":{"opacity":[0,1]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] [{"selector":"#anim-df5a24b6-4448-47b0-86fe-2fbd721412d0","keyframes":{"transform":["translate3d(-116.19048%, 0px, 0)","translate3d(0px, 0px, 0)"]},"delay":0,"duration":1200,"easing":"cubic-bezier(0.2, 0.6, 0.0, 1)","fill":"both"}] 2

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How to Find Areas Related to Circles and Squares?

From each corner of a square of side 4 cm, a quadrant of a circle of radius 1 cm is cut & also a circle of diameter 2 cm is cut as shown. Find the area of the remaining portion of the square.

The area of the remaining portion of the given image is 9.72 square centimetres. Let us see the step-by-step calculation.

Given that the square given is of a side length of 4 centimetres.

Area of square ABCD = (side length)

Area of square ABCD = 4 = 16 cm

2

2

2

It is also given that all 4 corners of the squares represent quadrants of circles with a radius of 1 cm.

i.e., each quadrant is a sector that makes an angle at the centre.

So, the area of all 4 quadrants can be calculated using the area of sector formula:

Area of sector =

Given θ =

Area of 4 quadrants = 4 × Area of sector

degrees and

r = 1 cm.

Area of 4 quadrants =

Also, there is a cut in the centre of the square which is a circle of radius 1 cm.

Area of circle =

So, the area of the remaining portion of the square is:

A = 16 − 3.14 − 3.14

A = (area of ABCD) − (area of 4 sectors) − (area of circle)

A = 9.72 cm

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