In geometry, an “altitude” is a line segment within a shape that defines its perpendicular height relative to a chosen base. This concept helps in understanding various geometric figures.
Defining an Altitude
An altitude is a line segment originating from a shape’s vertex and extending perpendicularly to the opposite side or the line containing it. It forms a right angle with the side it meets and represents the shortest distance from the vertex to that side, serving as the shape’s height. It is sometimes referred to simply as the height.
The side to which an altitude is drawn is called the base. An altitude always forms a right angle with the base, whether the base is a side of the figure or an extension of that side.
Altitudes in Common Shapes
The concept of an altitude is most frequently discussed in triangles, where every triangle possesses three altitudes, one from each vertex. For an acute triangle, where all angles are less than 90 degrees, all three altitudes lie entirely within the triangle’s boundaries.
In a right-angled triangle, which contains one 90-degree angle, two of the altitudes actually coincide with the legs (the sides forming the right angle) of the triangle. The third altitude, drawn from the vertex with the right angle to the hypotenuse, lies inside the triangle. For an obtuse triangle, characterized by one angle greater than 90 degrees, two of the altitudes fall outside the triangle. These altitudes are drawn from the acute angle vertices to the extended lines of the opposite sides, while the altitude from the obtuse angle vertex remains inside the triangle.
Beyond triangles, altitudes are also relevant for quadrilaterals, particularly parallelograms and trapezoids, though they are more commonly referred to as “height” in these figures. In a parallelogram, the height is the perpendicular distance between any pair of its parallel sides. A parallelogram has two distinct altitudes, corresponding to the two pairs of parallel sides. Similarly, for a trapezoid, the height (or altitude) is defined as the perpendicular distance between its two parallel bases.
Practical Uses of Altitudes
Altitudes serve a fundamental role in geometry, primarily as a component in calculating the area of various shapes. The length of an altitude, when combined with the length of its corresponding base, allows for straightforward area determination. For a triangle, the area is calculated using the formula: half of the product of its base and its altitude (Area = 0.5 × base × height). This formula highlights how the altitude represents the true perpendicular height needed for this calculation.
For parallelograms, the area is found by multiplying the length of its base by its corresponding altitude (Area = base × height). Here, the altitude provides the perpendicular separation between the chosen base and its opposite parallel side. In the case of a trapezoid, the area formula also incorporates its altitude. It is calculated as half the sum of the lengths of its two parallel bases multiplied by the altitude (Area = 0.5 × (base1 + base2) × height). In all these applications, the altitude provides the necessary perpendicular distance measurement, which is crucial for accurately determining the space enclosed by these two-dimensional figures.