Задание 3 · Стереометрия (краткий ответ)
3.5. Объем составного многогранника
Для репетитора: показать ответы
1. Формулы объёма
| Тело | Объём |
|---|---|
| Призма (любая) | \(V=S_{\text{осн}}\cdot h\) |
| Прямоугольный параллелепипед | \(V=abc\) |
| Куб | \(V=a^3\) |
| Пирамида | \(V=\dfrac13S_{\text{осн}}\cdot h\) |
Единицы: \(1\ \text{дм}^3=1000\ \text{см}^3\), \(1\ \text{м}^3=1000\ \text{дм}^3\), \(1\) литр \(=1\ \text{дм}^3\).
2. Два способа найти объём составного многогранника
Способ 1. Достроить до целого тела и вычесть лишнее
- Достройте фигуру до прямоугольного параллелепипеда или куба.
- Найдите объём этого целого тела.
- Найдите объём «лишней» части.
- Вычтите: \(V=V_{\text{целого}}-V_{\text{лишней}}\).
Пример схемы: параллелепипед \(10\times9\times4\) имеет объём \(360\); лишняя часть \(5\times4\times7\) имеет объём \(140\); тогда \(V=360-140=220\).
Способ 2. Разбить на части
- Разделите фигуру на параллелепипеды.
- Найдите объём каждого.
- Сложите результаты.
3. Фигура — призма с «ломаным» основанием
Если фигуру можно рассматривать как прямую призму, объём равен \(V=S_{\text{осн}}\cdot h\). Площадь основания находят, разбивая его на прямоугольники.
4. Из одинаковых кубиков и брусков
- Объём фигуры из одинаковых кубиков равен числу кубиков, умноженному на объём одного кубика.
- Если из параллелепипеда вынули один единичный кубик, объём уменьшается на объём этого кубика.
- Фигуру из одинаковых брусков сначала «соберите»: найдите измерения итогового параллелепипеда по рисунку, затем вычислите \(V=abc\).
5. Подобие
При увеличении всех линейных размеров в \(k\) раз объём увеличивается в \(k^3\) раз, а площадь поверхности — в \(k^2\) раз. Отношение объёмов подобных фигур равно кубу коэффициента подобия.
6. Если не хватает размера
- Недостающее ребро (высоту, ребро основания) находят по теореме Пифагора из диагоналей граней: например, по диагонали боковой грани и ребру основания.
- Если известен объём и два ребра параллелепипеда, третье ребро находят делением.
- При переливании жидкости объём сохраняется; при увеличении площади основания в \(n\) раз уровень падает в \(n\) раз.
Разобранные примеры
Пример 1
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Найдите объем пространственного креста, изображенного на рисунке и составленного из единичных кубов.
Решение
Крест состоит из 7 одинаковых кубов, поэтому его объем в 7 раз больше объема одного куба.
Ответ: 7.
Пример 2
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
Решение
Объем многогранника равен разности объемов куба, ребро которого равно 5, и прямоугольного параллелепипеда с ребрами 1, 3, 5, откуда находим: 53 − 1 · 3 · 5 = 110.
Ответ: 110.
Пример 3
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
Решение
Объем данного многогранника равен сумме объемов параллелепипедов с ребрами 5, 2, 1 и 1, 2, 2:
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Ответ: 14.
- «Многогранники», задание 3 ЕГЭ, examer.ru
- «Задание 3 профильного ЕГЭ: объёмы многогранников — формулы и методы решения», school-pro.ru
- «Геометрия. Многогранники», материалы 5.4–5.7, school.kco27.ru
- «Призма и параллелепипед» (математика, профильный уровень), olympmo.ru, 2020
- «Стереометрия», материалы подготовки к ЕГЭ, abitur.ivanovo.ac.ru, 2022
- У1
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы многогранника прямые). - У2
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У3
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У4
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У5
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У6
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У7
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У8
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У9 Найдите объем многогранника, изображенного на рисунке (все двугранные углы многогранника прямые).
- У10 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- У11 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- У12 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- У13
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У14
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У15
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы многогранника прямые). - У16
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У17
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У18
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У19
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - У20
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Найдите объём многогранника, изображённого на рисунке. Все двугранные углы многогранника прямые. - У21
- У22
- У23
- У24
- Д1
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д2
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д3
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д4
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д5 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д6
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д7
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д8
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Найдите объём многогранника, изображённого на рисунке (все двугранные углы прямые). - Д9Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д10 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д11 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д12 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д13 Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д14
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Найдите объём многогранника, изображённого на рисунке (все двугранные углы прямые). - Д15
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д16
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/%3E%3Cpath d='m 73.9,142.8 c 0.6,0 1.3,0 2.0,0 0,-1.3 0,-2.7 0,-4.1 1.3,-0.2 0.6,1.5 0.7,2.3 0,0.8 0,1.6 0,2.4 -2.1,0 -4.2,0 -6.4,0 0,-1.5 0,-3.1 0,-4.7 1.3,-0.2 0.6,1.5 0.7,2.3 0.1,0.5 -0.2,1.6 0.2,1.7 0.5,-0.1 1.6,0.2 1.8,-0.2 0,-1.3 0,-2.6 0,-3.9 1.3,-0.2 0.6,1.5 0.7,2.3 0,0.5 0,1.1 0,1.7 z' /%3E%3Cpath d='m 80.3,143.9 c 0.3,1.4 -3.0,1.7 -1.7,0.7 1.5,-0.2 1.0,-1.9 0.5,-2.8 -0.4,-1.0 -0.8,-2.0 -1.2,-3.1 1.2,-0.2 1.2,1.4 1.7,2.2 0.2,0.2 0.4,1.9 0.7,1.2 0.4,-1.1 0.9,-2.2 1.3,-3.4 1.4,-0.2 0.0,1.5 -0.0,2.3 -0.3,0.9 -0.7,1.9 -1.1,2.9 z' /%3E%3Cpath d='m 87.6,140.8 c -0.4,0.8 -1.9,0.1 -2.8,0.3 -2.0,-0.1 0.3,2.4 1.5,1.5 2.5,0.0 -1.4,1.6 -2.3,0.2 -1.9,-1.5 -0.3,-5.3 2.1,-4.4 0.9,0.2 1.4,1.2 1.4,2.2 z m -0.7,-0.2 c 0.6,-2.3 -4.0,-1.0 -2.1,0.0 0.7,0 1.4,-0.0 2.1,-0.0 z' /%3E%3Cpath d='m 88.9,143.4 c 0,-1.5 0,-3.1 0,-4.7 1.1,0 2.2,0 3.3,0 0.0,1.5 -3.3,-0.4 -2.5,1.8 0,0.9 0,1.9 0,2.9 -0.2,0 -0.5,0 -0.7,0 z' /%3E%3Cpath d='m 93.1,142.5 c 0.7,1.4 4.2,-0.8 2.1,-1.2 -0.5,-0.1 -2.2,0.3 -1.5,-0.6 1.0,0.1 3.1,0.0 1.4,-1.2 -0.6,-0.7 -2.8,0.8 -1.8,-0.6 2.0,-0.9 4.2,1.2 3.3,3.2 -0.2,1.5 -3.5,2.3 -3.6,0.7 l 0,-0.1 0,-0.0 0,0 z' /%3E%3Cpath d='m 98.4,142.3 c 1.6,-0.5 0.6,2.3 -0.0,0.6 -0.0,-0.2 0.0,-0.4 0.0,-0.6 z' /%3E%3Cpath d='m 101.8,142.7 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 107.1,141.0 c -0.8,1.9 2.7,2.3 2.0,0.1 0.6,-2.4 -2.3,-2.7 -2.0,-0.1 z m 2.0,4.2 c 0,-0.8 0,-1.6 0,-2.4 -1.8,2.2 -3.8,-1.7 -2.6,-3.3 0.4,-1.8 3.0,0.3 2.6,-1.1 -0.4,-1.1 0.9,-1.8 0.8,-0.3 -0.7,2.2 2.1,-0.6 2.4,1.4 1.2,1.6 -0.3,5.2 -2.4,3.5 -0.4,0.6 0.6,2.5 -0.8,2.1 z m 2.8,-4.2 c 0.8,-1.9 -2.7,-2.3 -2.0,-0.1 -0.6,2.4 2.3,2.7 2.0,0.1 z' /%3E%3C/g%3E%3C/svg%3E)
Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д17
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0,-3.1 0,-4.7 1.1,0 2.2,0 3.3,0 0.0,1.5 -3.3,-0.4 -2.5,1.8 0,0.9 0,1.9 0,2.9 -0.2,0 -0.5,0 -0.7,0 z' /%3E%3Cpath d='m 138.9,257.4 c 0.7,1.4 4.2,-0.8 2.1,-1.2 -0.5,-0.1 -2.2,0.3 -1.5,-0.6 1.0,0.1 3.1,0.0 1.4,-1.2 -0.6,-0.7 -2.8,0.8 -1.8,-0.6 2.0,-0.9 4.2,1.2 3.3,3.2 -0.2,1.5 -3.5,2.3 -3.6,0.7 l 0,-0.1 0,-0.0 0,0 z' /%3E%3Cpath d='m 144.2,257.2 c 1.6,-0.5 0.6,2.3 -0.0,0.6 -0.0,-0.2 0.0,-0.4 0.0,-0.6 z' /%3E%3Cpath d='m 147.6,257.6 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 152.9,255.9 c -0.8,1.9 2.7,2.3 2.0,0.1 0.6,-2.4 -2.3,-2.7 -2.0,-0.1 z m 2.0,4.2 c 0,-0.8 0,-1.6 0,-2.4 -1.8,2.2 -3.8,-1.7 -2.6,-3.3 0.4,-1.8 3.0,0.3 2.6,-1.1 -0.4,-1.1 0.9,-1.8 0.8,-0.3 -0.7,2.2 2.1,-0.6 2.4,1.4 1.2,1.6 -0.3,5.2 -2.4,3.5 -0.4,0.6 0.6,2.5 -0.8,2.1 z m 2.8,-4.2 c 0.8,-1.9 -2.7,-2.3 -2.0,-0.1 -0.6,2.4 2.3,2.7 2.0,0.1 z' /%3E%3C/g%3E%3C/svg%3E)
Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д18
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clip-path='url(%23SVGID_00000093142885361215633650000005274448106228908734_)'%3E%3Cg enable-background='new '%3E%3Cpath d='M38.2607,31.0835v1.0615h-1.3428v2.6758h-1.2173V32.145h-4.2349v-0.9561l4.6406-6.7119h0.8115v6.6064 H38.2607z M35.7006,31.0835v-5.0293l-3.5117,5.0293H35.7006z'/%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3C/g%3E%3Cg style='stroke:none;fill:%23000;fill-opacity:0.4' %3E%3Cpath d='m 124.4,156.1 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 133.3,154.2 c -0.4,0.8 -1.9,0.1 -2.8,0.3 -2.0,-0.1 0.3,2.4 1.5,1.5 2.5,0.0 -1.4,1.6 -2.3,0.2 -1.9,-1.5 -0.3,-5.3 2.1,-4.4 0.9,0.2 1.4,1.2 1.4,2.2 z m -0.7,-0.2 c 0.6,-2.3 -4.0,-1.0 -2.1,0.0 0.7,0 1.4,-0.0 2.1,-0.0 z' /%3E%3Cpath d='m 138.2,156.2 c 0.6,0 1.3,0 2.0,0 0,-1.3 0,-2.7 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Найдите объём многогранника, изображённого на рисунке. Все двугранные углы многогранника прямые. - Д19
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Найдите объем многогранника, изображенного на рисунке (все двугранные углы прямые). - Д20
- Д21
- Д22
- Д23
