Теперь вам не нужно тратить время на долгие вычисления, прежде чем вы сможете узнать площадь треугольника. Зная методы расчета, используемые для расчета площади треугольника, вы легко сможете это сделать самостоятельно. Действительно, всегда лучше знать формулы площади треугольника. Треугольники могут быть разными и вы это знаете, но как найти площадь треугольника если вам практически ничего неизвестно о треугольнике? И что нужно знать из размеров треугольника, чтобы найти его площадь. Давайте разбираться. При этом тема не так проста как кажется на первый взгляд, наверное, поэтому задачи нахождения площади треугольника есть и в ОГЭ и в ЕГЭ по математике.
Содержание
Что такое треугольник
Треугольник – это геометрическая фигура. По определению, это многоугольник, имеющий три стороны. Следовательно, треугольник также должен иметь три угла.
Сумма трех углов треугольника должна быть равна 180°.
Чтобы иметь возможность вычислить площадь треугольника, мы должны сначала знать меру его основания, а также высоту. Основание треугольника представляет одну из его сторон. Высота, с другой стороны, представляет собой каждую из трех прямых линий, которые проходят через одну из вершин треугольника и перпендикулярны стороне, лежащей напротив принятой вершины (то есть перпендикулярно основанию).
Прежде всего, помните, что треугольник состоит из трех сторон и трех углов. Это значит, что у него должно быть три вершины. Треугольник, вершинами которого являются A, B и C, может быть представлен как: ΔABC. Существуют разные виды треугольников. Они могут быть классифицированы двумя различными способами: либо по свойству его сторон, либо по свойству его углов.
Различные типы треугольников в зависимости от длины их сторон
Разносторонний треугольник
Мы узнаем разносторонний треугольник по трем сторонам, которые имеют разную длину. Эта треугольная форма может быть построена только с тремя разными углами. Кроме того, один из них может быть прямым углом (или углом 90 °). В общем, название «произвольный треугольник» используется для разностороннего треугольника.
Равнобедренный треугольник
Мы говорим, что треугольник равнобедренный, если он имеет две стороны одинаковой длины и два равных угла при основании. Равнобедренный треугольник также можно узнать по тому факту, что его высота представляет его ось симметрии, его медиану и биссектрису.
Прямоугольный треугольник
Прямоугольный треугольник обязательно имеет прямой угол. Другими словами, сумма двух других его углов должна быть равна 90°. Прямоугольный треугольник также имеет гипотенузу.
Это противоположная сторона вершине с прямым углом. Прямой треугольник может быть разносторонним (или любым), если его три стороны имеют разную длину.
Кроме того, он может быть равнобедренным в том случае, если он имеет два одинаковых катета.
Равносторонний треугольник
Треугольник называется равносторонним, если он имеет три стороны одинаковой длины. Поэтому все его углы также равны и каждый по 60°. В равностороннем треугольнике любая высота также выступает в качестве медианы и биссектрисы.
Площадь треугольника
Площадь разностороннего треугольника
Вычисляем площадь треугольника без особенностей – все его стороны разные и все углы разные.
Если известны две стороны треугольника и угол между ними, то площадь разностороннего треугольника вычисляется по формуле “площадь треугольника через две стороны и угол между ними”:
Если известны высота в треугольнике и основание, то используется формула площади треугольника через основание и высоту:
Формула Герона определения площади треугольника
Если известны стороны любого треугольника, то его площадь можно определить по формуле Герона.
, где
Площадь равнобедренного треугольника
Площадь треугольника через основание и сторону можно найти, если известны сторона и основания равнобедренного треугольника.
К равнобедренному треугольнику также применима формула площади треугольника через основание, сторону и угол между ними:
Найти площадь равнобедренного треугольника можно также через боковые стороны и угол между ними.
Площадь равнобедренного треугольника через основание и угол между боковыми сторонами:
Площадь прямоугольного треугольника
Приведем формулы площади прямоугольного треугольника. Формула площади прямоугольного треугольника через катет и прилежащий угол:
Площадь прямоугольного треугольника по радиусу вписанной окружности и гипотенузе
Площадь прямоугольного треугольника, если в него вписана окружность:
Площадь равностороннего треугольника
Площадь равностороннего треугольника можно найти через радиус описанной окружности.
Если дан радиус вписанной окружности, то площадь равностороннего треугольника можно найти по формуле:
Площадь равностороннего треугольника, если известна сторона треугольника:
Площадь равностороннего треугольника, если известна высота треугольника:
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