Triangle area:
half of base times perpendicular height

Why a triangle is half a rectangle

Draw a rectangle and cut it along a diagonal: you get two identical right triangles. Each is exactly half the rectangle, so a right triangle with legs 12 and 7 has half the area of a 12 by 7 rectangle. The same idea extends to triangles that lean. Any triangle can be fitted inside a rectangle whose width is the base and whose height is the triangle's height, and the triangle fills precisely half of it.

That is the source of the one rule you need, and it is why the answer to this class of question is always a half.

The formula

You can choose any side as the base, but the height must then be measured to that same base at 90 degrees. Three different base-and-height pairs on the same triangle give the same area, which is a useful check on measurements.

Worked example: base 12 cm, height 7 cm

If the units were metres the answer would be 42 m², and that is a good point to remember: halving does not change units, but multiplying two lengths gives square units. Reporting 42 cm rather than 42 cm² is a frequent slip.

1. 1Pick a side as the base and note its length.
2. 2Find the perpendicular height to that base, not the length of a slanted side.
3. 3Multiply base by height.
4. 4Halve the product and attach square units.

The height trap

The most common wrong answer comes from using a slanted side as the height. Take an isosceles triangle with two sides of 13 cm and a base of 10 cm. Using 13 as the height gives 10 × 13 ÷ 2 = 65 cm², which is too large. The true perpendicular height drops from the apex to the midpoint of the base, splitting the triangle into two right triangles with a hypotenuse of 13 and a short leg of 5. The height is therefore √(13² − 5²) = √144 = 12 cm, and the correct area is 10 × 12 ÷ 2 = 60 cm².

When the height lies outside the triangle

In an obtuse triangle the perpendicular from the apex to the base lands outside the base itself, on an extension of the base line. The rule still holds: take the base as the side you chose, drop a perpendicular to the extension of that line, and use that length. A site plan with a wedge-shaped plot is a typical case.

If the dimensions you have are three sides and no height, use Heron's formula: compute the semi-perimeter s = (a + b + c) ÷ 2, and area = √(s(s − a)(s − b)(s − c)). It reproduces 60 cm² for the 13, 13, 10 triangle: s = 18, and √(18 × 5 × 5 × 8) = √3,600 = 60.

Area of a triangular plot or panel in practice

Triangles turn up whenever a rectangle meets a diagonal. A gable end of a house that is 6 m wide at the eaves and rises 2.2 m to the ridge is a triangle with area 6 × 2.2 ÷ 2 = 6.6 m². If you are painting it, that is the surface to cover, and a one-litre tin that covers around 10 m² per coat will do a coat with a little left over.

A corner plot with a 30 m frontage along one road and a perpendicular depth of 18 m to the opposite corner has an area of 30 × 18 ÷ 2 = 270 m². Note that the height is taken from the apex to the road line, which is what a surveyor would call the offset.

When you know two sides and the angle between them, the area is ½ × a × b × sin(C). Two sides of 10 and 8 with a 30 degree angle between them give ½ × 10 × 8 × 0.5 = 20.

Checking the answer for sense

A triangle's area is always less than the rectangle that encloses it, so compare your answer with base × height. If your result is larger than that product, something is wrong. If it is exactly half, you have it. A second check uses similar triangles: doubling both base and height multiplies the area by four, not two, which is a useful reality check when scaling plans.

Finally, mind precision. Area values keep as many significant figures as the measurements justify. If the base is measured to the nearest 0.1 m and the height to the nearest 0.1 m, quoting an area to four decimal places suggests an accuracy that is not there. One or two decimals is plenty.

As a last exercise, find the area of a right triangle with legs of 9 m and 40 m. The two legs are perpendicular, so area = 9 × 40 ÷ 2 = 180 m². The hypotenuse is √(81 + 1,600) = 41 m, and using it as a base would require a height of 2 × 180 ÷ 41 = 8.78 m. The same triangle, three different base-and-height pairs, one area.

Practical uses

• Gable ends and triangular roof sections when you estimate paint, boards or cladding.
• Corner plots and triangular garden beds where a rectangle area tool will not apply directly.
• Cross-sections of embankments, where triangle areas are summed to get earthwork volumes.
• Sail areas, kite fabric and ramps in design drawings.

For anything with straight sides, break the shape into triangles and rectangles, find each area, and sum. A dedicated triangle area tool can confirm hand calculations quickly.