Base and Height of a Triangle
Learning to identify the base and height in different triangles
1. What is a Triangle?
A triangle has 3 sides and 3 angles.
Triangle ABC has three sides: AB, BC, and AC
2. What is the Base?
Key Fact:
The base of a triangle can be any of its three sides!
We usually choose one side to be the base, then find the height from that base.
Same Triangle, Different Bases:
Base at bottom
Base on left side
Base on right side
3. What is the Height?
Definition:
The height of a triangle is the perpendicular distance from the base to the opposite vertex (corner).
The height is always at a 90° angle (right angle) to the base.
Height from Vertex to Base:
EG is the base. FH is the height.
FH ⊥ EG (FH is perpendicular to EG)
⚠️ Important Rule:
The height of a triangle is always perpendicular to its base.
4. Height Inside or Outside the Triangle?
Sometimes the height falls inside the triangle, and sometimes it falls outside!
Height INSIDE
In acute-angled triangles (all three angles are smaller than 90°), the height is inside
Height OUTSIDE
In obtuse-angled triangles (one angle is greater than 90°), the height may be outside
5. Practice Questions
Practice 1:
In Triangle PQR, PQ is the base. Which line segment represents the height?
ShowHide answer
Answer: The height is RS because it is perpendicular to the base PQ.
Practice 2:
Look at this obtuse-angled triangle. If AB is the base, where does the height meet the line of the base?
ShowHide answer
Answer: Draw the height from C at right angles to AB. It does not meet AB itself: extend AB past B, and the height meets the extended base line at a point outside the triangle (call it D).
Practice 3:
Triangle XYZ is a right-angled triangle. Which two sides can be used as the base and the height?
ShowHide answer
Answer: In this right-angled triangle:
- If YZ is the base, then XY is the height
- If XY is the base, then YZ is the height
The two sides that form the right angle can be used as base and height!
Key Takeaways
- The base of a triangle can be any of its three sides
- The height is the perpendicular distance from the opposite vertex to the base
- The height is always at a 90° angle to the base
- For obtuse-angled triangles, the height may fall outside the triangle
- For right-angled triangles, the two sides forming the right angle can be the base and height