Finding the height of an obtuse triangle often feels counterintuitive at first. Mastering this concept is essential for geometry students, engineers, and anyone working with structural design. Unlike right triangles where a leg serves as the height, or acute triangles where all altitudes lie neatly inside the figure, an obtuse triangle forces you to look outside the box—literally. Even so, two of the three altitudes in an obtuse triangle fall completely outside the shape, extending from a vertex to the extension of the opposite side. This guide breaks down every method to calculate the height of an obtuse triangle, from basic area formulas to advanced trigonometry, ensuring you can tackle any problem with confidence.
Understanding the Geometry of an Obtuse Triangle
Before diving into calculations, it is vital to visualize what makes an obtuse triangle unique. By definition, an obtuse triangle has one interior angle measuring greater than 90° but less than 180°. The side opposite this obtuse angle is always the longest side of the triangle Easy to understand, harder to ignore..
The term "height" (or altitude) refers to the perpendicular distance from a vertex to the line containing the opposite side (the base). Think about it: in an acute triangle, all three altitudes intersect the opposite sides inside the triangle. Practically speaking, in a right triangle, two altitudes are the legs themselves. In an obtuse triangle, however, only the altitude dropped from the obtuse angle vertex lands on the opposite side inside the triangle. The altitudes dropped from the two acute vertices must extend outward to meet the extensions of their opposite sides.
Not the most exciting part, but easily the most useful.
This distinction is not just academic; it dictates how you label your diagram and which formulas you apply. If you are given a diagram, always extend the base line (using a dashed line) when drawing the height from an acute vertex.
Method 1: Using the Area Formula (Most Common Approach)
The most frequent scenario in textbook problems and real-world applications provides the area of the triangle and the length of the base. Since every triangle has three bases and three corresponding heights, you must identify which base the problem refers to Not complicated — just consistent..
Worth pausing on this one The details matter here..
The standard area formula is: $A = \frac{1}{2} \times b \times h$
Where:
- $A$ = Area
- $b$ = Length of the chosen base
- $h$ = Height (altitude) corresponding to that base
To find the height, simply rearrange the formula: $h = \frac{2A}{b}$
Step-by-Step Example
Problem: An obtuse triangle has an area of 60 square centimeters. The side chosen as the base measures 15 cm. Find the height relative to this base.
- Identify knowns: $A = 60 \text{ cm}^2$, $b = 15 \text{ cm}$.
- Apply formula: $h = \frac{2 \times 60}{15}$.
- Calculate: $h = \frac{120}{15} = 8 \text{ cm}$.
Crucial Note: This height (8 cm) is the perpendicular distance from the opposite vertex to the line containing the 15 cm base. If the 15 cm side is adjacent to the obtuse angle, this height falls inside the triangle. If the 15 cm side is opposite the obtuse angle (the longest side), the height from the obtuse vertex falls inside, but heights from the other vertices fall outside. The formula remains exactly the same regardless of where the altitude segment lands.
Method 2: Using Trigonometry (Two Sides and Included Angle)
Often, you are not given the area directly. Instead, you might know the lengths of two sides and the angle between them (SAS). Consider this: this is where trigonometry shines. You can calculate the area first, then derive the height, or solve for the height directly.
The SAS Area Formula
If you know sides $a$ and $b$ and the included angle $C$: $A = \frac{1}{2} ab \sin(C)$
Once you have the area, plug it into $h = \frac{2A}{\text{base}}$.
Direct Height Calculation via Sine
You can bypass the area step entirely if you know which height you need. Imagine you know side $a$ and side $b$, and the angle $C$ between them. You want the height ($h_a$) dropped onto side $a$ (the base).
The height forms a right triangle with side $b$ (acting as the hypotenuse of that right triangle) and the angle adjacent to the base Worth keeping that in mind..
- If the known angle $C$ is the angle between side $a$ and side $b$, then the height relative to base $a$ is: $h_a = b \sin(C)$
- Conversely, the height relative to base $b$ is: $h_b = a \sin(C)$
Wait—what if the known angle is obtuse? The sine of an obtuse angle ($\theta > 90^\circ$) is positive and equal to the sine of its supplement ($180^\circ - \theta$). Take this: $\sin(120^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$. The formula $h = b \sin(C)$ works perfectly even when $C$ is obtuse. The geometry handles the "outside" nature automatically because the sine function defines the vertical component of the side length.
Example: SAS with Obtuse Angle
Problem: Sides $a = 10$ and $b = 12$ form an obtuse angle $C = 120^\circ$. Find the height relative to base $a$.
- Identify: Base $= a = 10$. Side adjacent to base $= b = 12$. Included Angle $C = 120^\circ$.
- Formula: $h_a = b \sin(C)$.
- Calculate: $h_a = 12 \times \sin(120^\circ)$.
- Value: $\sin(120^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$.
- Result: $h_a = 12 \times 0.866 = 10.392$ units.
Method 3: Using Heron’s Formula (Three Sides Known - SSS)
When you only know the three side lengths (SSS), you cannot use simple trigonometry because you don't know any angles initially. Heron’s Formula allows you to find the area using only side lengths, after which you can find any height.
Heron’s Formula Steps
- Calculate the semi-perimeter ($s$): $s = \frac{a + b + c}{2}$
- Calculate the Area ($A$): $A = \sqrt{s(s-a)(s-b)(s-c)}$
- Calculate the desired height ($h$) relative to chosen base ($b_{base}$): $h = \frac{2A}{b_{base}}$
Example: SSS Obtuse Triangle
Problem: An obtuse triangle has sides $a = 13$, $b = 14$, $c = 20$. Find the height relative to the longest side ($c = 20$).
- Verify it is obtuse: Check if $c^2 > a^2 + b^2$. $20^2 = 400$. $13^2 + 14^2 = 169 + 196 = 365$. Since $400 > 365$, the angle opposite side $c$ is obtuse. The height from this vertex will fall inside
