Students often get confused when it comes to altitude in triangles because they just look at the longest side. A triangle has three sides therefore it has three bases. The altitude of this geometric shape is a perpendicular segment drawn from a vertex to the opposite. With this in mind, every triangle has three altitudes. Have you ever worked on how to determine the area of given triangles by themselves or as part of other shapes' by drawing the altitude and using the correct formula? These worksheets can be used to learn and practice finding the altitude and median of a triangle. This will also help you better understand the concept of the area of a triangle by drawing its altitude and then using the base x altitude formula.
Print Medians and Altitudes of Triangles Worksheets
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Medians and Altitudes of Triangles Lesson
This lesson will walk student through how to calculate the altitude and recognizing the point that would lead to a establishing a median line segment.
Practice Worksheet
Using the grid and following the steps below, draw the altitude of the given triangle and then find its area. We then cover 2 related practice problems.
Mega Altitude Worksheet
This worksheet allows you the opportunity to find a wide range of various altitudes of triangles. You will then have the ability to spot their medians for extra credit.
Drawing Altitude and Area Worksheet
For each triangle, draw its altitude and then find its area. You can also begin to draw medians.
Practice Worksheet
Draw the altitude of the triangle after you determine it. There are 8 triangles to work with in total.
What is a Median of a Triangle?
The median is a simple line segment that is drawn from a vertex (where two sides meet) to the midpoint of the opposite side. In the real-world medians are calculated by engineers to add structural reinforcement for building and all types of structures, since this is strongest point of the opposite base. In the diagram below, the red line segment serves as the median of this triangle.
How to Find the Altitude of Triangles
If you owned a triangle manufacturing company and had to deliver them to your customers in a rectangular box, how would you know what sized box to order? You would only know it once you know the triangle's height or altitude.
An altitude is a straight line from a triangle's vertex to the opposite side or base, keeping the constructed line perpendicular to the base. Each side of the triangle is called its base. The altitude of the triangle varies depending on the base you’re using for measurement. Hence, each triangle has 3 altitudes or 3 heights.
The altitude of a triangle, or height, is a line from a vertex to the opposite side that is perpendicular to that side. In other words, the distance from one side to the opposite vertex is called the altitude of this shape.
If we know the measurement of the three sides of a triangle, (ha, hb and hc), we can use the Heron’s formula for finding the three altitudes:
Here, a, b, and c are the three sides of the triangle, and s is the semi perimeter.
Formulas for Finding the Altitude of a Triangle
Even though the most common way to find the altitude of a triangle is through the length of the sides, other formulas exist that can help us find the altitude if the known and unknown variables vary.
1. Using the Area of a Triangle
The formula of finding the area of a triangle can also be used to find the altitude of a right-angled triangle.
Area = b x h / 2
Here, b stands for base and h shows the height.
So, h = 2 x area / b
2. Calculating Area Using Triangle Sides
Heron's formula is an equation that lets you calculate the area of the triangle if you have been the measurement of all the sides. After calculating the area, you can apply the basic formula to calculate the altitude of a triangle.
Heron's formula to calculate the area of a triangle goes like this:
area=0.25 x √((a + b + c)* (-a + b + c)* (a - b + c)* (a + b - c) )
3. Given Two Sides and the Angle Between
Start by finding out the area of the triangle using the following formulas, depending on which side of the triangle you have been given:
Area = 0.5 * a * b * sin(Υ)
Area = 0.5 * a * c * sin(Β)
Area = 0.5 * a * c * sin(Α)
Altitude or height h = 2 * 0.5 * a * b * sin(Υ) / b = a * sin(Υ)
How to Calculate the Height of an Equilateral Triangle
An equilateral triangle has three equal sides, and all of its angles equal 60°. Because all three sides have equal lengths, the altitude of all three sides will also be equal and can be calculated by applying the Pythagorean theorem:
h2 + (a/2)2 = a2
Here, a is any side of the triangle. And height (h) is (√3 a)/2
How to Calculate the Height of an Isosceles Triangle
A triangle with two sides of equal length and one side of different length is called an isosceles triangle.
h2 + (b/2)2 = a2
In this Pythagorean equation, h represents the altitude, and b/2 and h are the sides, where a is the hypotenuse.
How to Calculate the Height of a Right Triangle
A triangle with a 90° angle is called a right triangle. It is easy to calculate the altitude of 2 sides of a right triangle because the two sides are perpendicular to each other. If you consider the small side as a base, the longer side will be the altitude and vice versa. To calculate the third altitude, use the following formula:
hc = area x 2/c = a x b/c