The height in bh/2 is measured at a right angle to the base. Using the length of a slanted side instead is the commonest mistake on this topic, and it always gives an answer that is too large, because a slanted side is the hypotenuse of a right-angled triangle whose vertical leg is the real height. ...
The height has to be measured at a right angle to the base. Using the length of a slanted side instead is the commonest mistake here, and it always gives an answer that is too large. The animation below shows why: slide the apex sideways and the triangle changes shape completely while the area stays exactly the same — because neither the base nor the perpendicular height has moved. Four ways in are supported, and answers stay exact: a 2-3-4 triangle has area 3√15/4, and 2.9047 is a rounding of it.
Any one of the four descriptions. Whole numbers, decimals and fractions all work.
BASE
HEIGHT
AREA
30
Exactly 30 — this one comes out rational, which is less common than the textbook examples suggest.
THE ANSWER PUT BACK
from the drawn corners → 30 ✓
BASE × HEIGHT
60
the rectangle this is half of
HEIGHT ÷ BASE
0.6
how steep it is, whatever the size
DOUBLE EVERY LENGTH
120
four times the area, not twice
WHAT IS HAPPENING
10 × 6 = 60
That product is the area of a rectangle on the same base with the same height. The triangle is exactly half of it, which is what the halving in the next step is for.
A = 60 ÷ 2 = 30
Two copies of any triangle fit together into a parallelogram on the same base and of the same height, so one triangle is half of that. The height must be measured at a right angle to the base — the length of a slanted side is always longer and gives an answer that is too large.
Why only the perpendicular height counts. As the apex slides, the two slanted sides lengthen and shorten, the angles all change, and the triangle stops looking like the one it started as. What has not changed is the base it stands on or its distance above that base — and those are the only two numbers in bh/2. Measure the height along a slanted side instead and you are using the hypotenuse of a right-angled triangle whose vertical leg is the true height, so the number is always too big, and it gets worse the more the triangle leans.
Exact values throughout · surds kept as surds, never rounded away
Pick which description you have — base and height, three sides, two sides with the angle between them, or three corners — and fill in the boxes.
Read the decimal first, and the exact value beneath it. Where the area is a surd, that surd is the answer and the decimal is a rounding of it.
Press slide and watch the area readout. The apex moves, the shape changes entirely, and the number does not budge — that is what "perpendicular height" means in practice.
If three sides are refused, read which pair fell short. Two sides have to be longer together than the third, or they cannot reach across to meet above it.
Take a triangle with base 10 and height 6. Multiply: 10 × 6 = 60. That is the area of a rectangle on the same base with the same height. Halve it: A = 30. The halving is not a convention. Take a second copy of any triangle, turn it through 180° and set it against the first, and the two fit exactly into a parallelogram on the same base and of the same height. One triangle is half of that parallelogram, whatever its shape. Now the part people get wrong. Suppose the apex sits well off to one side, so the left-hand side of the triangle slants a long way. That slanted side might measure 11.7 while the height is 6. Using 11.7 gives an area of 58.5 instead of 30 — nearly double, and nothing about the number looks suspicious. The height must be the perpendicular distance, which is the shortest distance from the apex to the base line, not the distance along any side. Slide the apex and see it directly: with base 10, an apex at (1, 6) and an apex at (7, 6) describe very different-looking triangles and both have area 30, because both are 6 above the same base. Now three sides: 2, 3 and 4. Half the perimeter is s = (2 + 3 + 4) ÷ 2 = 4.5. Heron's formula multiplies four terms: 4.5 × (4.5 − 2) × (4.5 − 3) × (4.5 − 4) = 4.5 × 2.5 × 1.5 × 0.5 = 8.4375. The square root of that is the area, and 8.4375 is 135/16, so A = √(135/16) = 3√15/4 exactly, which is about 2.9047. Try 1, 2 and 10 instead and there is no triangle at all: 1 + 2 = 3, which is less than 10, so the two short sides cannot reach across to meet above the long one. Heron's formula catches it — the factor (s − 10) comes out negative — but only if you refuse to take the square root of a negative number rather than quietly taking its size. And three corners, say (1, 1), (5, 2) and (3, 7). The shoelace sum is 1(2 − 7) + 5(7 − 1) + 3(1 − 2) = −5 + 30 − 3 = 22, and the area is half of that: 11. The sign only records which way round the corners were listed, so it is dropped.
| Rule | What it says | Why |
|---|---|---|
| Base and height | A = bh ÷ 2 | The height is measured at a right angle to the base, never along a side. |
| Why the half | two copies make a parallelogram | Rotate one copy through 180° and the two fit exactly. |
| Sliding the apex | changes nothing | Move it parallel to the base and the area is identical. Only the perpendicular height counts. |
| Three sides | A = √(s(s−a)(s−b)(s−c)) | Heron’s formula, with s half the perimeter. |
| When three sides fail | a + b must exceed c | Otherwise the factors turn negative and no triangle exists. |
| Two sides and the angle | A = ½ab·sin C | The angle must be the one between the two sides, not any other. |
| Why that works | b·sin C is the height | It is still base times height over two, with the height found by trigonometry. |
| Three corners | A = ½|x₁(y₂−y₃) + …| | The shoelace sum. Its sign gives the direction round; the size gives the area. |
| A rational area is rare | 2.2% of whole-number triangles | Most areas are surds. 3-4-5 giving 6 is the exception, not the pattern. |
| Heron on thin triangles | breaks in floating point | The four factors nearly cancel. 98% of a needle-shaped test set came out wrong. |
| Doubling every length | multiplies the area by four | Two lengths are involved, so the scale factor is squared. |
| The tallest possible | a right angle between the sides | sin C peaks at 90°, so that is where two given sides enclose the most. |
| Equilateral of side a | A = √3 a² ÷ 4 | Never rational for a rational side, because √3 is irrational. |
| Units | lengths in cm give cm² | An area answer without a squared unit means the wrong formula was used. |
A = bh ÷ 2, where b is any side you choose as the base and h is the perpendicular distance from the opposite corner to that base. Any of the three sides can be the base, as long as the height is measured to the one you picked — the three pairings all give the same area.
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Last updated: August 30, 2026 · Exact values throughout, surds kept as surds · The height is measured at a right angle to the base, never along a side.