An altitude of a triangle is a line segment drawn from one vertex straight down to the opposite side, meeting that side at a perfect 90-degree angle. Constructing one with a compass and straightedge is a core geometry skill, and the process is really just a specific application of a broader construction: dropping a perpendicular from a point to a line. Here’s how to do it, along with shortcuts for special triangles and an alternative method if you have a set square.
What You Need
The classic construction requires only a compass and a straightedge (an unmarked ruler). You’ll also want a sharp pencil and enough room on your paper, since some steps involve arcs that swing wide of the triangle itself. If your class allows other tools, a set square and ruler offer a faster path covered below.
Compass and Straightedge: Step by Step
Suppose you have triangle ABC and you want to construct the altitude from vertex A down to side BC. The goal is to draw a line from A that hits BC (or BC extended) at exactly 90 degrees.
Step 1: Extend the Base
Use your straightedge to extend side BC in both directions, well past the triangle. This matters because in obtuse triangles the altitude’s foot lands outside the original side. Extending the line now prevents problems later.
Step 2: Set Your Compass Width
Place the compass point on vertex A. Open it wide enough so that when you draw an arc, it crosses the extended line BC in two places. Swing the arc. Label the two intersection points P and Q.
Step 3: Find the Perpendicular Foot
Now move the compass point to P. Open it to a radius greater than half the distance from P to Q (eyeballing this is fine, just make it generous). Draw an arc below the line. Without changing the compass width, move the point to Q and draw another arc that crosses the first one. Label the crossing point R.
Step 4: Draw the Altitude
Use your straightedge to draw a line through vertex A and point R. This line is perpendicular to BC. The segment from A to the point where it meets line BC is the altitude. Mark the right angle at the base with a small square to show it’s 90 degrees, and label the foot of the altitude (often called D).
That’s the full construction. You can repeat the same process from vertex B to side AC, or from vertex C to side AB, to construct the other two altitudes of the triangle.
Why Extending the Base Matters
In an acute triangle (all angles less than 90 degrees), every altitude lands inside the triangle, so extending the base isn’t strictly necessary. In an obtuse triangle, though, the altitude from at least one vertex falls outside the triangle. If you don’t extend the opposite side into a full line first, your arcs won’t have anything to intersect. Getting into the habit of extending the base every time keeps the construction reliable regardless of triangle type.
Altitudes in Right Triangles
Right triangles have a useful shortcut. The two legs of the triangle already meet at 90 degrees, which means each leg is itself an altitude. The leg from one acute vertex meets the other leg at the right angle vertex perpendicularly by definition, so two of the three altitudes are already drawn for you.
The third altitude, from the right angle vertex to the hypotenuse, is the interesting one. Constructing it follows the same compass and straightedge steps above: treat the hypotenuse as your base line, place the compass on the right angle vertex, and drop the perpendicular. This altitude divides the hypotenuse into two segments and creates two smaller right triangles inside the original. A useful property called the Right Triangle Altitude Theorem says the length of this altitude is the geometric mean of those two hypotenuse segments. In practical terms, if the altitude splits the hypotenuse into segments of length a and b, then the altitude’s length equals the square root of a times b.
Using a Set Square and Ruler
If your assignment doesn’t require a compass-only construction, a set square (the right-angled drafting triangle) paired with a ruler is faster. To draw the altitude from vertex A to side BC:
- Align the ruler along side BC so it sits flat against the base.
- Place the set square on the ruler so one short edge sits flush against the ruler’s edge.
- Slide the set square along the ruler until the vertical edge touches vertex A.
- Draw the line along that vertical edge from A down to BC. Mark the right angle and label the foot point D.
The set square guarantees a 90-degree angle because its edges are manufactured at exactly that angle. This method is common in drafting and in classes that focus on practical geometry rather than formal constructions.
Where All Three Altitudes Meet
Every triangle has three altitudes, one from each vertex. A key theorem in geometry states that all three altitudes of any triangle intersect at a single point called the orthocenter. In an acute triangle the orthocenter sits inside the triangle. In an obtuse triangle it falls outside, beyond the longest side. In a right triangle it lands exactly on the right angle vertex, since the two legs (which are altitudes themselves) already meet there.
Finding the orthocenter is a common geometry exercise. You only need to construct two of the three altitudes; the point where they cross is the orthocenter, and the third altitude will pass through the same point. This is a good way to check your work: if the third altitude doesn’t hit the intersection, one of your constructions is slightly off.
Tips for Cleaner Constructions
Keep your compass pencil sharp. Thick arcs make it hard to pinpoint exact intersections, and even a millimeter of error at the crossing point can tilt the altitude off true perpendicular. When drawing the final line through A and R, press lightly with the straightedge first and check that the line looks perpendicular before committing to a dark line. If the triangle is small on your paper, widen the compass more than you think you need; larger arcs produce intersection points that are farther apart and easier to identify precisely.