{"id":7209,"date":"2021-12-30T07:11:21","date_gmt":"2021-12-30T07:11:21","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=7209"},"modified":"2022-01-03T06:58:23","modified_gmt":"2022-01-03T06:58:23","slug":"surface-area-of-frustum-with-examples-and-faqs","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/surface-area-of-frustum-with-examples-and-faqs\/","title":{"rendered":"Surface Area of Frustum with Examples and FAQs"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
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Surface Area of Frustum with Examples and FAQs<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

What Is the Surface Area of Frustum?<\/b><\/h2>\n

<\/b><\/p>\n

\"\"<\/b><\/p>\n

 <\/p>\n

The surface area is of two types:
<\/b><\/h3>\n

(1) <\/strong>Curved Surface Area<\/b>: <\/span>The curved surface area of a\u00a0frustum of a cone is the area surrounded by its curved face<\/span>. In other words, it\u2019s the area that excludes the area of the circular top and bottom surfaces.<\/span><\/p>\n

<\/span><\/p>\n

(2)<\/strong> <\/span>Total Surface Area<\/b>: <\/span>The total surface area of a frustum of a cone is the sum of the areas enclosed by all its faces.<\/span><\/p>\n

<\/span><\/p>\n

Formula for Surface Area of Frustum<\/b><\/h2>\n

\n

\n

\"\"<\/p>\n

\n

The figure represents a frustum, where h is its height, l is the slant height and r_{1} \\text { and } r_{2} \\text { (where } r_{1}>r_{2} \\text { ) }<\/span><\/span>\u00a0the radii of the lower and upper circular bases respectively, of the frustum of a cone.<\/span><\/p>\n

<\/b>\\text { Slant height, l }=\\sqrt{h^{2}+\\left(r_{1}-r_{2}\\right)^{2}}<\/span><\/p>\n

The formula for the Curved Surface Area (C.S.A.) of the Frustum is:<\/span><\/p>\n

\\text { C.S.A }=\\pi \\times\\left(\\mathbf{r}_{1}+\\mathbf{r}_{2}\\right) \\times \\mathbf{l}<\/span><\/span><\/p>\n

The formula for the Total Surface Area(T.S.A.) of the Frustum is:<\/span><\/p>\n

\\text { T.S.A. } =\\text { C.S.A. }+\\text { Area of the upper face }+\\text { Area of the base } <\/span><\/span><\/p>\n


\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\pi\\left(r_{1}+r_{2}\\right) {\\text{l}}+\\pi r_{1}^{2}+\\pi r_{2}^{2}<\/span><\/span><\/p>\n

\u00a0The value of \u03c0 is taken to be \\frac{22}{7} \\text { or } 3.14<\/span>.<\/span><\/p>\n

 <\/p>\n

Examples<\/b><\/h2>\n

<\/b><\/p>\n

Example 1<\/strong>:\u00a0<\/b>Calculate the curved surface area and total surface area of the frustum of a right circular cone of height 12 cm, large base radius to be 20 cm, and smaller base radius to be 15 cm. Find the surface area in terms of \u03c0 only.<\/span><\/p>\n

Solution:<\/strong>\u00a0<\/span><\/p>\n

The height of the frustum of the cone, h = 12 cm.<\/span><\/p>\n

The large base radius of the frustum,r_{1}=20 \\mathrm{~cm}<\/span><\/span>.<\/span><\/p>\n

The small base radius of the frustum, r_{2}=15\\mathrm{~cm}<\/span><\/span>.<\/span><\/p>\n

We know that,<\/span><\/p>\n

\\text { Slant height, } =\\sqrt{h^{2}+\\left(r_{1}-r_{2}\\right)^{2}} <\/span><\/span><\/p>\n


\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\sqrt{12^{2}+(20-15)^{2}} \\mathrm{~cm} <\/span><\/span><\/p>\n


\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=\\sqrt{144+(5)^{2}} \\mathrm{~cm}=\\sqrt{169} \\mathrm{~cm}=13 \\mathrm{~cm}\n<\/span>
<\/span><\/p>\n

Thus, the curved surface area of the given frustum of the right circular\u00a0cone is,<\/span><\/p>\n

\\text { C.S.A. } =\\pi \\times\\left(r_{1}+r_{2}\\right) \\times l <\/span><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=\\pi \\times(20+15) \\times 13 \\mathrm{~cm}^{2}<\/span><\/span><\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=455 \\pi \\mathrm{cm}^{2}<\/span><\/span><\/p>\n

And, the total surface area of the frustum is,<\/span><\/p>\n

\\text { T.S.A. } =\\pi\\left(r_{1}+r_{2}\\right) l+\\pi r_{1}^{2}+\\pi r_{2}^{2} <\/span><\/span><\/p>\n


\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\left(455 \\pi+\\pi \\times 20^{2}+\\pi \\times 15^{2}\\right) \\mathrm{cm}^{2} <\/span><\/span><\/p>\n


\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=(455 \\pi+400 \\pi+225 \\pi) \\text{ }\\mathrm{cm}^{2} <\/span><\/span><\/p>\n


\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=1080 \\pi \\text{ } \\mathrm{cm}^{2}<\/span><\/span><\/p>\n

Hence the curved surface area of the frustum of the right circular cone is 455 \\pi \\mathrm{cm}^{2}<\/span><\/span> and its total surface area is 1080\\pi \\text{\u00a0 } \\mathrm{cm}^{2}<\/span> <\/span>.<\/span><\/p>\n

<\/span><\/p>\n

Example 2<\/strong>:<\/strong> If <\/span>a cone is cut by a plane horizontally, we get a frustum. The radii of the circular top and base of the frustum are 9 m and 4 m, respectively. The slant height of the frustum is 10 m. Then find the curved surface area of the frustum taking the value of <\/span>\u03c0 as 3.14.<\/span><\/p>\n

Solution<\/strong>:<\/strong>\u00a0<\/span><\/p>\n

The large base radius of the frustum,r_{1}=9 m .<\/span><\/span><\/p>\n

The small base radius of the frustum, r_{2}=4m .<\/span><\/span><\/p>\n

Slant height, l = 10 m<\/span><\/p>\n\\therefore \\text { C.S.A. } =\\pi \\times(9+4) \\times 10 \\mathrm{~m}^{2} <\/span>\n=130 \\pi \\mathrm{m}^{2} <\/span>\n=130 \\times 3.14 \\mathrm{~m}^{2} \\quad[\\text { taking } \\pi=3.14] <\/span>\n=408.2 \\mathrm{~m}^{2}<\/span>\n

\\text{Hence the curved surface area of the frustum of the cone is } 408.2 \\mathrm{~m}^{2}<\/span>.<\/p>\n

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Explore Other Topics<\/h1>\n

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Geometry<\/a><\/div>\n
Trigonometry<\/a><\/div>\n
Operations<\/a><\/div>\n
Numbers<\/a><\/div>\n<\/div>\n

[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

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\n
Mensuration Formula<\/a><\/div>\n
Surface Area and Volume Formula<\/a><\/div>\n
Surface Area of Cone<\/a><\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1. What is a frustum of a cone?
    <\/strong><\/h3>\n

    Ans: <\/strong>When a plane slices a cone parallel to its base then the lower part of the cone is known as a frustum and the upper part retains to be a cone.
    <\/span><\/span><\/p>\n

    Q2. What is meant by the Surface Area of Frustum?
    <\/strong><\/h3>\n

    Ans: <\/strong>The surface area of the frustum is the sum of the area enclosed by all its faces. There are two types of surface area (1) Curved Surface Area and (2) Total Surface Area.
    <\/strong><\/p>\n

    Q3. What is the formula for the slant height of the frustum of a cone?<\/strong><\/h3>\n

    Ans:<\/strong> \\text { Slant height }=\\sqrt{\\text { height }^{2}+(\\text { radius of larger base }-\\text { radius of smaller base })^{2}}<\/span><\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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