Ever wondered what the altitude of a triangle actually is (and why test-makers love to sneak it into questions on exams like the SAT, ACT, or even GRE)? Let’s break it down in a simple, no-nonsense way! The altitude of a triangle is just a fancy name for a line segment that starts at one vertex (corner) of the triangle and drops straight down—at a perfect 90° angle—to the side opposite that vertex. Think of it like a height line, standing tall and proud!

Now, here’s something cool: depending on the type of triangle you’re dealing with, the altitude might fall neatly inside the triangle, or it might stretch out and land outside the triangle entirely. Yep, it’s one of those geometry quirks that keeps you on your toes during tests!

Also, different triangles have slightly different altitude situations. For example:

• In acute triangles, all altitudes fall inside.
• In right triangles, one altitude is literally one of the sides.
• And in obtuse triangles, you’ll find the altitudes stretching outside the triangle.

Mastering how altitudes behave isn’t just for passing a geometry class—it’s a game-changer for scoring higher on standardized tests like PSAT/NMSQT, AP exams, or even the GMAT and MCAT. Knowing where the altitude lands can save you time (and points!) on tricky geometry questions.

Altitude of a Triangle

The altitude of a triangle is defined as the perpendicular line segment drawn from the one vertex of a triangle to the side opposite to that particular vertex. The altitude makes a right angle i.e., \( 90\degree \) with the base of the triangle that it touches. It is commonly referred to as the height of a triangle and is represented by the letter h. It is measured by calculating the distance between the vertex and its opposite side. As there are three vertices, thus three altitudes can be drawn in every triangle from each of those vertices.

In the above image \( \triangle ABC \) is a triangle where AD is the altitude from vertex A on side BC, BE is another altitude from vertex B on side AC, and CF is the third altitude from vertex C on side AB.

We have different altitudes for different types of triangles. We can find the latitude of different types of triangle using the following formula table.

Altitude of Types of Triangle

The altitudes of triangles vary depending upon the type of triangles as mentioned below.

Altitude of Scalene Triangle

If all the three sides of a triangle have different length or if none of the sides of the triangle is equal to each other, then the triangle is known as a scalene triangle.

In the above image \( \triangle ABC \) is a scalene triangle with AD as the altitude.

We can find the length of the altitude using the following formula.

\( {\rm{h}} = \frac{{2\sqrt {s(s – a)(s – b)(s – c)} }}{{{\rm{ base }}}} \)

Where, h is the altitude length,

s = semi perimeter,

a, b, and c are length of the three sides, and

Base is the length of the base.

Altitude of Isosceles Triangle

If any two of the three sides of a triangle are equal then the triangle is known as the isosceles triangle. The altitude of an isosceles triangle bisects the apex angle and also bisects the base. Thus, the altitude of the isosceles triangle divides the triangle into two congruent triangles by SSS axiom.

In the above image \( \triangle ABC \) is an isosceles triangle with AD as the altitude.

We can find the length of the altitude using the following formula.

\( h=\sqrt{a^2-\frac{b^2}{4}} \),

Where, h is the altitude length,

a = length of the congruent sides, and

b = base length.

Altitude of Equilateral Triangle

If all the three sides of a triangle have the same length then the triangle is referred to as an equilateral triangle. The altitude of an equilateral triangle bisects the base and the opposite angle of the base. Thus, the altitude of the isosceles triangle divides the triangle into two congruent triangles by SSS axiom.

In the above image \( \triangle ABC \) is an equilateral triangle with BD as the altitude.

We can find the length of the altitude using the following formula.

\( h=\frac{\sqrt{3}}{2}a \)

Where, h is the altitude length, and

a is the length of the side.

Altitude of Right Triangle

In a right-angled triangle the perpendicular and the base are considered as altitudes. The altitude from the vertex at the right angle to the hypotenuse divides the triangle into two similar triangles.

In the above image \( \triangle ABC \) is a right triangle with CD as the altitude.

We can find the length of the altitude using the following formula.

\( h=\sqrt{xy} \)

Where, h=height, and

x and y are the length of the line segments formed by the altitude drawn on the hypotenuse.

Altitude of Obtuse Triangle

If any one angle in a triangle is an obtuse-angle, then the triangle is called an obtuse-angled triangle. An angle whose measure is more than \( 90\degree \) is called an obtuse-angled triangle. The other two angles are acute i.e., less than \( 90\degree \).

The altitude lies outside the triangle for an obtuse-angled triangle. In this type of triangle the base is stretched and then a perpendicular is constructed from the opposite vertex to the base which is the altitude.

In the above image \( \triangle ABC \) is a right triangle with AD as the altitude.

Intersection of Altitudes of a Triangle

A triangle has three altitudes which arise from its three vertices. These three altitudes intersect at a point known as the Orthocentre of the triangle. Orthocenter does not need to lie inside the triangle only, it may also lie outside of the triangle.

In the above image, \( \triangle ABC \) has the Orthocentre at H which is the intersection of the altitudes AD, BE, and CF.

The orthocenter varies for different types of triangles.

• In an acute angled triangle, orthocenter lies inside the triangle.
• In an obtuse angled triangle, orthocenter lies outside the triangle.
• In a right triangle, orthocenter lies on the vertex of the right angle i.e., at the intersection of the base and perpendicular as the base and perpendicular are themselves the other two altitudes.

In the above image,

\( \triangle ABC \) is an acute angled triangle which has its orthocentre inside the triangle at H,

\( \triangle PQR \) is a obtuse angled triangle which has its orthocentre outside the triangle at R, and

\( \triangle XYZ \) is a right angled triangle which has its orthocentre on the trignel at X.

Characteristics of Altitude of a Triangle

The properties of Altitude of a Triangle are listed below.

• There can be maximum of three altitudes for any triangle
• The altitude of a triangle must be perpendicular to the opposite side i.e., it forms \( 90\degree \) with the opposite side.
• The altitude can lie inside or outside the triangle depending on the type of that particular triangle.
• The point of intersection of three altitudes is known as the orthocenter of the triangle.

Altitude of a Triangle Solved Example

Problem 1:The area of a right angled triangle is given as \( 81cm^2 \). Find the length of the altitude if the length of the base is 9 cm.

Solution:

We know the formula to find the area of a right angle triangle i.e., \( A=\frac{1}{2}\times b\times h \), where A is the area, b is base length, and h is altitude length.

Now we have been given the area and base length i.e., \( 81cm^2 \) and \( b=9cm \).

Therefore, \( A=\frac{1}{2}\times b\times h\Rightarrow h=\frac{2\times A}{b}=\frac{2\times81}{9}=18cm \), which is the required altitude length.

Problem 2:Calculate the altitude length of an isosceles triangle whose base length is 3cm and congruent sides have length of 5cm.

Solution:

\( h=\sqrt{a^2-\frac{b^2}{4}} \),

Where, h is the altitude length,

a is the length of the congruent sides, and

b is the base length.

We have been given a=5cm and b=3cm.

Therefore, \( h=\sqrt{a^2-\frac{b^2}{4}}=\sqrt{\left(5\right)^2-\frac{\left(3\right)^2}{4}}=\sqrt{25-\frac{9}{4}}=\sqrt{\frac{91}{4}}=\frac{\sqrt{91}}{2}cm \), which is required altitude length.

Conclusion

And there you have it! The altitude of a triangle isn’t just some boring geometry term—it’s a powerful tool that can pop up in all kinds of standardized tests like the SAT, ACT, GRE, or AP exams. Whether it’s inside, outside, or smack on the triangle, knowing how to find and use altitudes can seriously boost your problem-solving skills and save you precious time on test day!