KS3 · Maths
Area of a triangle
A tall, neat triangle and one leaning so far it looks about to topple, on the same base. Which covers more space? Drag the corner and find out.
Maths · Geometry
Same base + same perpendicular height = same area, however far C leans.
C can only slide along the faint line, which runs parallel to the base AB. The upright line on the right shows the perpendicular height: the gap between the base line and C's line, measured at a right angle. Drag C and watch the chips.
Where the half comes from
Take any triangle and make an exact copy. Turn the copy upside down (a half turn) and slide it against one side of the original, so the two fit together with no gaps.
Which one is the height?
Commit first. This is the decision that makes or breaks every triangle area question.
Triangle PQR has its base QR = 6 cm along the bottom. The top corner P leans out past R. PR = 5 cm is a slanted side. A dashed line PS = 4 cm drops straight down from P and meets the line of the base, extended beyond R, at a right-angle mark. S is outside the triangle. For base QR, which length is the perpendicular height?
Worked example: pick the right pair
Problem
A triangle rests on its longest side, which is 10 cm. Its other two sides are 6 cm and 8 cm, and they meet at the top corner with a right-angle mark. A dashed line 4.8 cm drops from that top corner to the 10 cm side, also with a right-angle mark. Find the area of the triangle.
WHAT YOU'VE LEARNED
A quick recap of today's lesson.
What you need to know
- Area of a triangle = ½ × base × perpendicular height, which is the same as (base × perpendicular height) ÷ 2.
- Any side can be the base. The perpendicular height is the distance from the base to the opposite corner, measured at a right angle to the base.
- The height can lie outside the triangle when the top corner overhangs the base. Extend the base line and measure to it.
- A slanted side is not the height. Only use a line as the height if a right-angle mark shows it meets the base at 90°.
- Two identical triangles make a parallelogram with the same base and height, which is why the formula halves.
- Area needs a base and its perpendicular height; perimeter needs all three sides.
The big picture
The area of any triangle is half of base × perpendicular height. You can choose any side as the base. The perpendicular height is the distance from that base to the opposite corner, measured at a right angle, and it may fall outside the triangle. The half is there because two copies of the triangle make a parallelogram with the same base and height.
Key points
Worked example
Problem
A triangle has a base of 2p cm and a perpendicular height of 9 cm. One of its slanted sides is marked q cm. Write an expression for its area.
⚠ Watch out
Using a slanted side as the height, or forgetting to halve. The height must meet the base at a right angle, and base × height gives the parallelogram, so the triangle is only half of it.
Memory hook
Base times height, then cut it in half: every triangle is half a parallelogram. And the height always stands up straight at a right angle, never leaning like a slanted side.
Check yourself
Base 21 cm, perpendicular height 12 cm, slanted sides 13 cm and 20 cm. Which two lengths do you need, and what is the area? (21 cm and 12 cm: 126 cm².)
Flashcards
(13)Area of a triangle: what do you multiply, and what do you do next?
Which side of a triangle is the base?
What is the perpendicular height of a triangle?
Can the perpendicular height be outside the triangle?
Can a slanted side be used as the height?
A line looks upright but has no right-angle mark. Can you use it as the height?
Why does the triangle formula include a half?
Two triangles have the same base and the same perpendicular height but look very different. How do their areas compare?
A diagram marks four or five lengths. Which do you use for the area?
How do you write the area of a triangle with base b and perpendicular height h?
What do you need to find the perimeter of a triangle, compared with its area?
You know a triangle's area and its base. How do you find its perpendicular height?
What units is the area of a triangle measured in?
Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.
Learning with Lightbulb is opening soon
You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.
Keep me postedMore KS3 Maths topics
- Adding and subtracting positive and negative integers
- Algebraic notation and conventions
- Angle sum in a triangle
- Angles on parallel lines (corresponding, alternate, co-interior)
- Area of a circle
- Area of a trapezium
- Area of composite rectilinear shapes
- Arithmetic sequences and finding the nth term
- Associative, commutative and distributive laws
- Bisecting an angle
- Calculating theoretical probabilities
- Changing the subject of a formula
How this lesson was checked. This KS3 Mathslesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 30 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.