document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170768>Question 1170768</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A regular pentagonal pyramid has an altitude of 20 cm and a slant height which measures 25cm. Find the base and the lateral areas of the pyramid. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851102 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851102\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step to find the base and lateral areas of the pentagonal pyramid.\r<BR>\n" );
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document.write( "**1. Understand the Geometry**\r<BR>\n" );
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document.write( "* **Regular Pentagonal Pyramid:**<BR>\n" );
document.write( "    * Base is a regular pentagon (all sides and angles equal).<BR>\n" );
document.write( "    * Altitude (height) is the perpendicular distance from the apex to the center of the base.<BR>\n" );
document.write( "    * Slant height is the altitude of a lateral face (a triangle).\r<BR>\n" );
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document.write( "**2. Visualize the Right Triangle**\r<BR>\n" );
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document.write( "Imagine a right triangle formed by:\r<BR>\n" );
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document.write( "* The altitude (20 cm)<BR>\n" );
document.write( "* The slant height (25 cm)<BR>\n" );
document.write( "* The apothem (a) of the pentagon (distance from the center of the pentagon to the midpoint of a side).\r<BR>\n" );
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document.write( "Using the Pythagorean theorem:\r<BR>\n" );
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document.write( "* a² + altitude² = slant height²<BR>\n" );
document.write( "* a² + 20² = 25²<BR>\n" );
document.write( "* a² + 400 = 625<BR>\n" );
document.write( "* a² = 225<BR>\n" );
document.write( "* a = √225 = 15 cm\r<BR>\n" );
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document.write( "**3. Find the Side Length of the Pentagon (s)**\r<BR>\n" );
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document.write( "* Let's consider a right triangle formed by:<BR>\n" );
document.write( "    * The apothem (a = 15 cm)<BR>\n" );
document.write( "    * Half of a side (s/2)<BR>\n" );
document.write( "    * The radius (r) of the pentagon.\r<BR>\n" );
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document.write( "* The central angle of a regular pentagon is 360° / 5 = 72°.<BR>\n" );
document.write( "* The angle formed by the apothem and the radius is half of the central angle, which is 36°.\r<BR>\n" );
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document.write( "* Using trigonometry (tangent):<BR>\n" );
document.write( "    * tan(36°) = (s/2) / a<BR>\n" );
document.write( "    * s/2 = a * tan(36°)<BR>\n" );
document.write( "    * s/2 = 15 * tan(36°)<BR>\n" );
document.write( "    * s/2 ≈ 15 * 0.7265<BR>\n" );
document.write( "    * s/2 ≈ 10.8975<BR>\n" );
document.write( "    * s ≈ 21.795 cm\r<BR>\n" );
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document.write( "**4. Find the Base Area (B)**\r<BR>\n" );
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document.write( "* Area of a regular pentagon:<BR>\n" );
document.write( "    * B = (1/2) * apothem * perimeter<BR>\n" );
document.write( "    * Perimeter (P) = 5 * side length<BR>\n" );
document.write( "    * P = 5 * 21.795 ≈ 108.975 cm<BR>\n" );
document.write( "    * B = (1/2) * 15 cm * 108.975 cm<BR>\n" );
document.write( "    * B ≈ 817.3125 cm²\r<BR>\n" );
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document.write( "**5. Find the Lateral Area (LA)**\r<BR>\n" );
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document.write( "* Lateral Area = (1/2) * perimeter * slant height<BR>\n" );
document.write( "* LA = (1/2) * 108.975 cm * 25 cm<BR>\n" );
document.write( "* LA ≈ 1362.1875 cm²\r<BR>\n" );
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document.write( "**Rounded Values:**\r<BR>\n" );
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document.write( "* Base Area (B) ≈ 817.31 cm²<BR>\n" );
document.write( "* Lateral Area (LA) ≈ 1362.19 cm²\r<BR>\n" );
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document.write( "**Final Answers:**\r<BR>\n" );
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document.write( "* **Base Area:** Approximately 817.31 cm²<BR>\n" );
document.write( "* **Lateral Area:** Approximately 1362.19 cm²<BR>\n" );
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