{"id":337,"date":"2022-04-13T10:43:42","date_gmt":"2022-04-13T10:43:42","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=337"},"modified":"2025-01-05T07:09:32","modified_gmt":"2025-01-05T07:09:32","slug":"r-4-area-of-sectors","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/r-4-area-of-sectors\/","title":{"rendered":"R.4 Area of sectors"},"content":{"rendered":"\n

Area of sectors<\/strong><\/h2>\n\n\n\n

Key Notes :<\/p>\n\n\n\n

A sector<\/strong> of a circle is a region bounded by an arc of the circle and the two radii that intersect the arc’s endpoints.<\/p>\n\n\n

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\"A<\/figure><\/div>\n\n\n

Here is a formula for the area of a sector:<\/strong><\/p>\n\n\n\n

K = m \/ 360 . A<\/p>\n\n\n\n

In this formula, K is the area of the sector, m is the degree measure of the arc bounding the sector (or the central angle that intercepts the arc), and A is the area<\/a> of the circle.<\/p>\n\n\n\n

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\n

Tip<\/strong><\/p>\n\n\n\n

You can also write this formula as a proportion<\/a> where each ratio relates the sector to the full circle:<\/p>\n\n\n\n

K \/ A = M \/ 360<\/p>\n\n\n\n

The ratio K \/ A compares the area of the sector to the area of the circle. The ratio m \/ 360<\/p>\n\n\n\n

compares the degree measure of the arc bounding the sector to the degree measure of a full circle.<\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n

Finding area of sectors<\/h2>\n\n\n\n

Let’s try it! Find the area of shaded sector BCD.<\/p>\n\n\n

\n
\"Circle<\/figure><\/div>\n\n\n

You can use the measure of BD  and the area of \u2a00C to find the area of shaded sector BCD. Find the area of the circle using the formula A = \u200b\ud835\udf0br2 , where r is the radius.<\/p>\n\n\n\n

A= \u200b\ud835\udf0br2<\/sup><\/p>\n\n\n\n

= \u200b\ud835\udf0b52<\/sup> Plug in r=5 .<\/p>\n\n\n\n

= 25\u200b\ud835\udf0b Simplify.<\/p>\n\n\n\n

The area of the circle is 25\u200b\ud835\udf0b square inches.<\/p>\n\n\n\n

Now, find the area of the sector.<\/p>\n\n\n\n

K = M \/ 360 . A<\/p>\n\n\n\n

K = 72 \/ 360 . 25\ud835\udf0b Plug in m=72 and A=25\u200b\ud835\udf0b.<\/p>\n\n\n\n

K = 5\u200b\ud835\udf0b Simplify.<\/p>\n\n\n\n

So, the area of shaded sector BCD is 5\u200b\ud835\udf0b square inches.<\/p>\n\n\n\n

Finding arc measures<\/h2>\n\n\n\n

To find the measure of an arc, you can use the area of the related sector and the area of the circle.<\/p>\n\n\n\n

Let’s try it! Find the measure, in degrees, of FIH.<\/p>\n\n\n

\n
\"Circle<\/figure><\/div>\n\n\n

FIH is the arc that bounds shaded sector FGH. So, you can use the area of shaded sector FGH and the area of \u2a00G to find the measure of the arc. The area of the shaded sector is 80\u200b\ud835\udf0b square centimeters. To find the area of the circle, you can use the formula A=\u200b\ud835\udf0br2, where r is the radius.<\/p>\n\n\n\n

A=\u200b\ud835\udf0br2<\/sup><\/p>\n\n\n\n

= \u200b\ud835\udf0b102<\/sup> Plug in r=10.<\/p>\n\n\n\n

= 100\u200b\ud835\udf0b Simplify.<\/p>\n\n\n\n

The area of the circle is 100\u200b\ud835\udf0b square centimeters.<\/p>\n\n\n\n

Now, find the measure of the arc.<\/p>\n\n\n\n

K = M \/ 360 . A<\/p>\n\n\n\n

80\u200b\ud835\udf0b = M \/ 360 . 100\ud835\udf0b Plug in K=80\u200b\ud835\udf0b and A=100\u200b\ud835\udf0b .<\/p>\n\n\n\n

28,800\u200b\ud835\udf0b \/ 100\u200b\ud835\udf0b = m Multiply both sides by <\/em>360 \/ 100\u200b\ud835\udf0b.<\/p>\n\n\n\n

288= m Simplify.<\/p>\n\n\n\n

So, the measure of FIH is 288\u00b0.<\/p>\n\n\n\n

Area of sectors and radians<\/h1>\n\n\n\n

You can also find the area of a sector when the arc measure is given in radians<\/a>. Here is a formula for the area of a sector, where K is the area of the sector, r is the radius of the circle, and \u200b\ud835\udf03 is the radian measure of the arc bounding the sector:<\/p>\n\n\n\n

K=<\/strong> 1\/2 r2\u200b<\/strong><\/sup>\ud835\udf03<\/strong><\/p>\n\n\n\n

Finding area of sectors<\/h2>\n\n\n\n

Let’s try it! Find the area of shaded sector XYZ.<\/p>\n\n\n

\n
\"Circle<\/figure><\/div>\n\n\n

Since you know the measure of \u2220XYZ in radians, you can use the radius of \u2a00Y and the measure of XZ to find the area of shaded sector XYZ. The measure of XZ is equal to the measure of the central angle that intercepts it. So, the measure of XZ is 2\u200b\ud835\udf0b3 radians.<\/p>\n\n\n\n

K=<\/strong> 1\/2 r2\u200b<\/strong><\/sup>\ud835\udf03<\/strong><\/p>\n\n\n\n

= 1\/2 . 62<\/sup> . 2\u200b\ud835\udf0b \/ 3 Plug in r=6 and \u200b\ud835\udf03= 2\u200b\ud835\udf0b \/ 3<\/p>\n\n\n\n

=12\u200b\ud835\udf0b Simplify.<\/p>\n\n\n\n

So, the area of shaded sector XYZ is 12\u200b\ud835\udf0b square meters.<\/p>\n\n\n\n

Finding arc measures<\/h2>\n\n\n\n

To find the measure of an arc in radians, you can use the area of the related sector and the radius of the circle.<\/p>\n\n\n\n

Let’s try it! Find the measure, in radians, of SU.<\/p>\n\n\n

\n
\"Circle<\/figure><\/div>\n\n\n

SU is the arc that bounds shaded sector STU. So, you can use the area of shaded sector STU and the radius of \u2a00T to find the measure of the arc. The area of the shaded sector is 6\u200b\ud835\udf0b square feet. You’re given the diameter of the circle, 8 feet, which is 2 times the length of the radius. So, the radius of the circle is 4 feet.<\/p>\n\n\n\n

K=<\/strong> 1\/2 r2\u200b<\/strong><\/sup>\ud835\udf03<\/strong><\/p>\n\n\n\n

6 \ud835\udf0b = 1\/2 . 42<\/sup> . \u200b\ud835\udf03 Plug in K=6\u200b\ud835\udf0b and r=4<\/p>\n\n\n\n

6\u200b\ud835\udf0b=8\u200b\ud835\udf03 Multiply.<\/p>\n\n\n\n

6\u200b\ud835\udf0b \/ 8 = \ud835\udf03 Divide both sides by 8.<\/p>\n\n\n\n

3\u200b\ud835\udf0b \/ 4 = \ud835\udf03 Simplify.<\/p>\n\n\n\n

So, the measure of SU is 3\u200b\ud835\udf0b \/ 4 radians.<\/p>\n\n\n\n

Learn with an example<\/strong><\/p>\n\n\n\n

\n
\n

The radius of a circle is 10 centimetres. What is the area of a sector bounded by a 180\u00b0 arc?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

Give the exact answer in simplest form.<\/p>\n\n\n\n

___________ square centimetres<\/p>\n<\/div><\/div>\n\n\n\n

The sector’s area depends on the arc’s measure and the circle’s area. You already know that the arc’s measure is 180\u00b0, so find the circle’s area.<\/p>\n\n\n\n

First, find the area of the circle.<\/p>\n\n\n\n

A = \ud835\udf0b r2<\/sup><\/p>\n\n\n\n

= \ud835\udf0b 102<\/sup> Plug in r=10<\/p>\n\n\n\n

= 100 \ud835\udf0b Square<\/p>\n\n\n\n

The area of the circle is 100\u200b\ud835\udf0b square centimetres.<\/p>\n\n\n\n

Now, find the area of the sector.<\/p>\n\n\n\n

K = A . m\/360<\/p>\n\n\n\n

= 100 \ud835\udf0b 180 \/ 360 Plug in A=100\u200b\ud835\udf0b and m=180<\/p>\n\n\n\n

= 50 \ud835\udf0b Multiply and simplify<\/p>\n\n\n\n

The area of the sector is  50\u200b\ud835\udf0b square centimetres.<\/p>\n<\/div><\/div>\n\n\n\n

\n
\n

The radius of a circle is 8 kilometres. What is the area of a sector bounded by a 45\u00b0 arc?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

Give the exact answer in simplest form.<\/p>\n\n\n\n

_________ square kilometres<\/p>\n<\/div><\/div>\n\n\n\n

The sector’s area depends on the arc’s measure and the circle’s area. You already know that the arc’s measure is 45\u00b0, so find the circle’s area.<\/p>\n\n\n\n

First, find the area of the circle.<\/p>\n\n\n\n

A = \ud835\udf0b r2<\/sup><\/p>\n\n\n\n

= \ud835\udf0b 82<\/sup> Plug in r=8<\/p>\n\n\n\n

= 64 \ud835\udf0b Square<\/p>\n\n\n\n

The area of the circle is 64\u200b\ud835\udf0b square kilometres.<\/p>\n\n\n\n

Now, find the area of the sector.<\/p>\n\n\n\n

K = A . m\/360<\/p>\n\n\n\n

= 64 \ud835\udf0b 45 \/ 360 Plug in A=64\u200b\ud835\udf0b and m=45<\/p>\n\n\n\n

= 8 \ud835\udf0b Multiply and simplify<\/p>\n\n\n\n

The area of the sector is  8\u200b\ud835\udf0b square kilometres.<\/p>\n<\/div><\/div>\n\n\n\n

\n
\n

The radius of a circle is 6 centimetres. What is the area of a sector bounded by a 90\u00b0 arc?<\/strong><\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

Give the exact answer in simplest form.<\/p>\n\n\n\n

___________ square centimetres<\/p>\n<\/div><\/div>\n\n\n\n

The sector’s area depends on the arc’s measure and the circle’s area. You already know that the arc’s measure is 90\u00b0, so find the circle’s area.<\/p>\n\n\n\n

First, find the area of the circle.<\/p>\n\n\n\n

A = \ud835\udf0b r2<\/sup><\/p>\n\n\n\n

= \ud835\udf0b 62<\/sup> Plug in r=6<\/p>\n\n\n\n

= 36 \ud835\udf0b Square<\/p>\n\n\n\n

The area of the circle is 36\u200b\ud835\udf0b square centimetres.<\/p>\n\n\n\n

Now, find the area of the sector.<\/p>\n\n\n\n

K = A . m\/360<\/p>\n\n\n\n

= 36 \ud835\udf0b 90 \/ 360 Plug in A=36\u200b \ud835\udf0b and m=90<\/p>\n\n\n\n

= 9 \ud835\udf0b Multiply and simplify<\/p>\n\n\n\n

The area of the sector is  9\u200b\ud835\udf0b square centimetres.<\/p>\n<\/div><\/div>\n\n\n\n

Let’s practice!<\/p>\n\n\n\n

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\"\"<\/a><\/figure>\n<\/div>\n\n\n\n
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\"\"<\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Area of sectors Key Notes : A sector of a circle is a region bounded by an arc of the circle and the two radii that intersect the arc’s endpoints. Here is a formula for the area of a sector: K = m \/ 360 . A In this formula, K is the area of the sector, m is the degree measure of theContinue reading “R.4 Area of sectors”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-337","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=337"}],"version-history":[{"count":21,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/337\/revisions"}],"predecessor-version":[{"id":17366,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/337\/revisions\/17366"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}