{"id":61941,"date":"2021-06-01T12:16:55","date_gmt":"2021-06-01T10:16:55","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61941"},"modified":"2026-09-30T20:37:55","modified_gmt":"2026-09-30T17:37:55","slug":"linear-size","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/linear-size\/","title":{"rendered":"Linear Size Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Find Size From Distance and Angle<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61941\" data-favorite-title=\"Linear Size\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/linear-size\/\" data-favorite-excerpt=\"Enter an object&#039;s angular size in degrees and its distance from the observer, and this Linear Size Calculator returns the object&#039;s actual linear size D using D = 2\u00b7L\u00b7tan(\u03b1\/2). Useful for astronomy, optics, photography, and surveying.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-geometry\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/linear-size\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61941\" data-post=\"61941\"><fieldset class=\"fieldset-input is-2\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Angular size (\u00b0)<\/label><input type=\"number\" name=\"angular_size_\" id=\"field-1\" placeholder=\"30\" step=\"0.01\" value=\"30\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Distance to the object (L)<\/label><input type=\"number\" name=\"distance_to_the_object_l\" id=\"field-2\" placeholder=\"10\" step=\"0.01\" value=\"10\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61941\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Linear size (D)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"61941\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">2<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Linear Size Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ D = 2 \\cdot L \\cdot \\tan\\left(\\frac{\\alpha}{2}\\right) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(D\\) \u2014 linear size of the object (the dimension perpendicular to the line of sight)<\/li>\n<li>\\(L\\) \u2014 distance from the observer to the object<\/li>\n<li>\\(\\alpha\\) \u2014 angular size subtended by the object, in degrees or radians<\/li>\n<\/ul>\n<p>The formula comes from simple isosceles triangle geometry: the observer, one edge of the object, and the opposite edge form a triangle that the distance line bisects, so half the linear size equals \\(L \\tan(\\alpha\/2)\\). Note that \\(\\alpha\\) must be converted to radians before applying standard tangent functions.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Linear Size \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a distant object is 10 units away and subtends an angular size of 30\u00b0. What is its linear size?<\/p>\n<p>\\[ D = 2 \\cdot 10 \\cdot \\tan\\left(\\frac{30^\\circ}{2}\\right) = 20 \\cdot \\tan(15^\\circ) \\approx 20 \\cdot 0.26795 \\approx 5.359 \\]<\/p>\n<p><strong>Result:<\/strong> the linear size is \\(D \\approx 5.3590\\) units \u2014 for example, an object 10 m away that spans 30\u00b0 of your view is about 5.36 m wide.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Linear Size Calculator<\/h2>\n<ol>\n<li>Enter the <strong>Angular size (\u00b0)<\/strong> \u2014 the angle the object occupies in your field of view, measured in degrees.<\/li>\n<li>Enter the <strong>Distance to the object (L)<\/strong> \u2014 the straight-line distance from the observer to the object, in any unit of length (meters, kilometers, feet, miles).<\/li>\n<li>Read the <strong>Linear size (D)<\/strong> \u2014 the result appears in the same unit of length you used for the distance.<\/li>\n<\/ol>\n<p>Use decimal degrees for the angular size; for instance, enter 0.5\u00b0 instead of 30 arcminutes. The result is rounded to four decimal places.<\/p>\n<h2>What Is Linear Size?<\/h2>\n<p>Linear size is the actual physical dimension of an object \u2014 its true width, height, or diameter \u2014 as opposed to its angular size, which is how large it <em>appears<\/em> from a particular viewpoint. The same object looks smaller as it moves farther away: the Sun and the Moon both subtend roughly 0.5\u00b0 in our sky, yet the Sun is about 400 times wider than the Moon because it is about 400 times farther away.<\/p>\n<p>The relationship between the two is governed by triangle geometry. If you draw lines from your eye to the two edges of an object, you form an isosceles triangle whose apex angle is the angular size \\(\\alpha\\) and whose equal sides have length \\(L\\). Bisecting the apex angle splits the object's size into two right triangles, each with a far side of \\(L \\tan(\\alpha\/2)\\), which gives the formula \\(D = 2L\\tan(\\alpha\/2)\\).<\/p>\n<p>This calculation appears everywhere angular measurements are made: astronomers use it to compute the diameters of planets, nebulae, and galaxies from their apparent sizes; surveyors use it to measure inaccessible objects such as cliffs or towers; photographers use it to check whether a subject will fill the frame; and optometrists and display engineers rely on it when working with field of view and screen viewing angles.<\/p>\n<p>To interpret the result, keep in mind that \\(D\\) is the dimension perpendicular to the line of sight. An object that is tilted or extended along the viewing direction will have other dimensions this formula cannot recover \u2014 it gives one linear measurement from one angular measurement and one distance.<\/p>\n<h2>Assumptions, Limitations, and Tips<\/h2>\n<ul>\n<li><strong>Flat geometry:<\/strong> the formula assumes straight-line (Euclidean) geometry, which is excellent for everyday distances but must be treated carefully for very large astronomical scales where curvature matters.<\/li>\n<li><strong>Small-angle approximation:<\/strong> for small angles (below about 10\u00b0), \\(D \\approx L \\alpha\\) with \\(\\alpha\\) in radians is a good shortcut. This calculator uses the exact tangent formula, so it stays accurate even for large angles.<\/li>\n<li><strong>Accurate inputs matter:<\/strong> the distance \\(L\\) should be measured to the object's center along the line of sight; the result inherits any error in both the angle and the distance.<\/li>\n<li><strong>Maximum angle:<\/strong> the formula is meaningful for angular sizes between 0\u00b0 and 180\u00b0. At \\(\\alpha = 180^\\circ\\) the tangent is undefined, and angles above that have no physical meaning in this context.<\/li>\n<li><strong>Units:<\/strong> the output has the same length unit as the distance input \u2014 there is no built-in unit conversion.<\/li>\n<\/ul>\n<p>If you need to work in the opposite direction, our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/angular-size\/\">Angular Size Calculator<\/a> finds the apparent angle from a known size and distance, and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/distance-perspective-projection\/\">Distance Perspective Projection Calculator<\/a> handles related projection problems.<\/p>\n<h2>Quick Reference Table<\/h2>\n<p>Linear size for a distance of \\(L = 100\\) at various angular sizes:<\/p>\n<table>\n<thead>\n<tr>\n<th>Angular size \u03b1<\/th>\n<th>Linear size D<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0.5\u00b0<\/td>\n<td>0.873<\/td>\n<\/tr>\n<tr>\n<td>1\u00b0<\/td>\n<td>1.745<\/td>\n<\/tr>\n<tr>\n<td>5\u00b0<\/td>\n<td>8.749<\/td>\n<\/tr>\n<tr>\n<td>10\u00b0<\/td>\n<td>17.633<\/td>\n<\/tr>\n<tr>\n<td>30\u00b0<\/td>\n<td>53.590<\/td>\n<\/tr>\n<tr>\n<td>45\u00b0<\/td>\n<td>82.843<\/td>\n<\/tr>\n<tr>\n<td>90\u00b0<\/td>\n<td>200.000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice how the linear size grows faster than the angle: doubling the angular size more than doubles \\(D\\) once the angle becomes large, because \\(\\tan\\) is nonlinear.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter an object's angular size in degrees and its distance from the observer, and this Linear Size Calculator returns the object's actual linear size D using D = 2\u00b7L\u00b7tan(\u03b1\/2). Useful for astronomy, optics, photography, and surveying.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[23],"class_list":["post-61941","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-geometry"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61941","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61941"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61941\/revisions"}],"predecessor-version":[{"id":66256,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61941\/revisions\/66256"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61941"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61941"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61941"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}