Triangle base, height and area
Find a perpendicular height and connect triangle area with half a parallelogram.
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The height meets the base at a right angle
Choose one side of a triangle as the base. Its height is the perpendicular distance from the opposite vertex to the line containing that base. The sloping side is usually not that height.
Base AB and its perpendicular height
A perpendicular from the opposite vertex meets the chosen base inside the triangle.
View figure data
| Point or marked feature | x (grid steps) | y (grid steps) |
|---|---|---|
| A | 0 | 5 |
| B | 8 | 5 |
| C | 2 | 0 |
| Line h start | 2 | 0 |
| Line h end | 2 | 5 |
| Right-angle mark 1 vertex | 2 | 5 |
| Right-angle mark 1 first arm | 8 | 5 |
| Right-angle mark 1 second arm | 2 | 0 |
Make a second copy of the triangle, turn it and join it to form a parallelogram with the same base and perpendicular height. Cut off its sloping end and move that piece to the other end: it fills a rectangle with the same base and height. Moving a piece preserves area, so the pair has area base × height. The two equal triangles share that area equally. Therefore
For base 8 cm and height 5 cm, area is 1/2 × 8 × 5 = 20 cm². The answer is in square centimetres. A base of 8 cm alone is not enough to find the area.
Heights can lie outside the triangle
If the opposite vertex lies beyond one end of your chosen base, extend the base line to draw the perpendicular height. The same rule applies. Rotating the page does not change the area or the right angle needed for the height.
The height reaches the extended base line
The perpendicular meets an extension of the base; the triangle boundary stays the same.
View figure data
| Point or marked feature | x (grid steps) | y (grid steps) |
|---|---|---|
| A | 0 | 5 |
| B | 8 | 5 |
| C | -2 | 0 |
| Line h start | -2 | 0 |
| Line h end | -2 | 5 |
| Line e start | 0 | 5 |
| Line e end | -2 | 5 |
| Right-angle mark 1 vertex | -2 | 5 |
| Right-angle mark 1 first arm | 0 | 5 |
| Right-angle mark 1 second arm | -2 | 0 |
This triangle also has base 8 cm and perpendicular height 5 cm, so its area is still 20 cm². The dashed extension is not an extra side of the triangle.
Check your understanding 1 Choose a height for a turned triangle
Use AB as the base. Name a side that is its perpendicular height, then give the base and height lengths. Each grid step is 1 cm.
A vertical base
AB is vertical and AC is horizontal. Use the right angle at A to pair the base and height.
View figure data
| Point or marked feature | x (grid steps) | y (grid steps) |
|---|---|---|
| A | 0 | 0 |
| B | 0 | 6 |
| C | 8 | 0 |
| Right-angle mark 1 vertex | 0 | 0 |
| Right-angle mark 1 first arm | 0 | 6 |
| Right-angle mark 1 second arm | 8 | 0 |
Show answer
AC (or CA) is perpendicular to AB. The base AB is 6 cm long and its height AC is 8 cm long. The sloping side BC is not perpendicular to AB. Turning the triangle does not change which lengths belong together.
Combine figures without counting an overlap twice
A rectangle with a triangular corner removed
The outer boundary follows five sides after a triangular corner is cut from a rectangle.
View figure data
| Point or marked feature | x (grid steps) | y (grid steps) |
|---|---|---|
| A | 0 | 0 |
| B | 8 | 0 |
| C | 12 | 3 |
| D | 12 | 7 |
| E | 0 | 7 |
A rectangle is 12 cm by 7 cm. A triangular corner with base 4 cm and perpendicular height 3 cm is removed. The remaining area is 12 × 7-1/2 × 4 × 3 = 84-6 = 78 cm².
Area counts square units; perimeter follows the exposed boundary. Here AB is 8 cm, BC is supplied as 5 cm, CD is 4 cm, DE is 12 cm and EA is 7 cm. The perimeter is 8 + 5 + 4 + 12 + 7 = 36 cm. The old rectangle’s perimeter was 38 cm: removing lengths of 4 cm and 3 cm and adding the new 5 cm edge gives the same 38-4-3 + 5 = 36 cm.
The removed triangle’s area does not tell you the new sloping side length. Use a supplied length for that side; do not count it as a horizontal or vertical grid distance.
Check your understanding 2 Keep area and perimeter separate
A 9 cm by 6 cm rectangle loses a triangular corner with base 3 cm and perpendicular height 4 cm. The new sloping boundary is supplied as 5 cm. Find the remaining area and perimeter, with units.
Show answer
The area is 9 × 6-3 × 4÷2 = 54-6 = 48 cm². The rectangle’s original perimeter was 2 × (9 + 6) = 30 cm. Replace the removed 3 cm and 4 cm lengths with the new 5 cm side: 30-3-4 + 5 = 28 cm. The triangular cut changes both quantities, but they count different things.
Check your understanding 3 Which given length belongs in the formula?
A triangle has base 10 cm, perpendicular height 6 cm and a sloping side 8 cm. Find its area. A learner uses 10 × 8÷2: explain the error.
Show answer
The area is 10 × 6÷2 = 30 cm². The 8 cm side is not perpendicular to the chosen base. Multiplying it by that base does not measure the enclosing parallelogram’s area.
Work backwards
A triangle has area 42 cm² and base 12 cm. Find its perpendicular height.
Show answer
Twice the area is 84 cm². Divide by the 12 cm base: height is 7 cm. Check 1/2 × 12 × 7 = 42 cm².
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