{"id":227531,"date":"2025-06-06T07:43:54","date_gmt":"2025-06-06T07:43:54","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=227531"},"modified":"2025-06-06T07:43:56","modified_gmt":"2025-06-06T07:43:56","slug":"a-triangle-is-inside-a-circle-where-the-triangles-base-is-on-the-circles-diameter-as-shown","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/06\/a-triangle-is-inside-a-circle-where-the-triangles-base-is-on-the-circles-diameter-as-shown\/","title":{"rendered":"A triangle is inside a circle where the triangle’s base is on the circle’s diameter as shown"},"content":{"rendered":"\n

A triangle is inside a circle where the triangle’s base is on the circle’s diameter as shown. What is the area of the shaded region? Use 3.14 for
. Enter your answer as a decimal in the box. ft
16 ft<\/p>\n\n\n\n

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

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Step 1: Understand the given dimensions<\/strong><\/h3>\n\n\n\n
    \n
  • The diameter of the circle is 16 ft<\/strong>.<\/li>\n\n\n\n
  • Since the triangle’s base is on the circle’s diameter and the triangle is inscribed, it forms a right triangle (by the Thales’ Theorem<\/strong>).<\/li>\n\n\n\n
  • The base of the triangle = diameter = 16 ft<\/strong><\/li>\n\n\n\n
  • Since it is an isosceles right triangle (as shown), the two equal legs are from the ends of the diameter up to the top of the circle, forming the height.<\/li>\n<\/ul>\n\n\n\n

    To find the height<\/strong> of the triangle, note that the radius = half of diameter = 8 ft, and the triangle is right-angled with the right angle at the top. So the height = radius = 8 ft<\/strong><\/p>\n\n\n\n

    \n\n\n\n

    Step 2: Area of the Circle<\/strong><\/h3>\n\n\n\n

    Areacircle=\u03c0r2=3.14\u00d7(8)2=3.14\u00d764=200.96 ft2\\text{Area}_{\\text{circle}} = \\pi r^2 = 3.14 \\times (8)^2 = 3.14 \\times 64 = 200.96 \\text{ ft}^2Areacircle\u200b=\u03c0r2=3.14\u00d7(8)2=3.14\u00d764=200.96 ft2<\/p>\n\n\n\n

    \n\n\n\n

    Step 3: Area of the Triangle<\/strong><\/h3>\n\n\n\n

    Areatriangle=12\u00d7base\u00d7height=12\u00d716\u00d78=64 ft2\\text{Area}_{\\text{triangle}} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = \\frac{1}{2} \\times 16 \\times 8 = 64 \\text{ ft}^2Areatriangle\u200b=21\u200b\u00d7base\u00d7height=21\u200b\u00d716\u00d78=64 ft2<\/p>\n\n\n\n

    \n\n\n\n

    Step 4: Area of the Shaded Region<\/strong><\/h3>\n\n\n\n

    Shaded Area=Areacircle\u2212Areatriangle=200.96\u221264=136.96 ft2\\text{Shaded Area} = \\text{Area}_{\\text{circle}} – \\text{Area}_{\\text{triangle}} = 200.96 – 64 = \\boxed{136.96 \\text{ ft}^2}Shaded Area=Areacircle\u200b\u2212Areatriangle\u200b=200.96\u221264=136.96 ft2\u200b<\/p>\n\n\n\n

    \n\n\n\n

    Explanation<\/strong><\/h3>\n\n\n\n

    In this problem, a triangle is inscribed in a circle with its base along the diameter. This is a classic geometric configuration where the triangle is a right triangle<\/strong> because of Thales’ Theorem<\/strong>, which states that any triangle inscribed in a circle with the diameter as one side is a right triangle. The right angle is opposite the diameter, meaning the triangle\u2019s height is the radius<\/strong> of the circle.<\/p>\n\n\n\n

    We are asked to find the area of the shaded region, which is the part of the circle not occupied by the triangle. This involves two key geometric formulas: the area of a circle<\/strong> and the area of a triangle<\/strong>. The radius of the circle is half the diameter, so here it is 8 ft. Using the formula for the area of a circle A=\u03c0r2A = \\pi r^2A=\u03c0r2, and substituting r=8r = 8r=8 and \u03c0=3.14\\pi = 3.14\u03c0=3.14, the total circle area is 200.96 ft\u00b2.<\/p>\n\n\n\n

    The triangle, being a right triangle with base 16 ft and height 8 ft, has an area of A=12\u00d7base\u00d7height=64A = \\frac{1}{2} \\times \\text{base} \\times \\text{height} = 64A=21\u200b\u00d7base\u00d7height=64 ft\u00b2.<\/p>\n\n\n\n

    To get the shaded area, we subtract the triangle\u2019s area from the circle\u2019s area: 200.96\u221264=136.96 ft2200.96 – 64 = 136.96 \\text{ ft}^2200.96\u221264=136.96 ft2<\/p>\n\n\n\n

    This result represents the area outside the triangle but still within the circle. This problem demonstrates the importance of understanding basic geometric properties and relationships in composite shapes.<\/p>\n\n\n\n

    \n\n\n\n

    \u2705 Final Answer: 136.96 ft\u00b2<\/strong><\/h3>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    A triangle is inside a circle where the triangle’s base is on the circle’s diameter as shown. What is the area of the shaded region? Use 3.14 for. Enter your answer as a decimal in the box. ft16 ft The Correct Answer and Explanation is: Step 1: Understand the given dimensions To find the height […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-227531","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227531","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=227531"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/227531\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=227531"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=227531"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=227531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}