How to Find the Area of an Equilateral Triangle From Its Side

Geometry Lessons

The short answer

A = (√3 / 4) × s²

Square the side length, multiply by the square root of three, and divide by four. A side of 10 gives (1.7321 / 4) × 100 = 43.30 square units. The formula is exact, and it is nothing more than half base times height with the height pre computed.

The equilateral triangle is the one shape where a single measurement settles everything. All three sides match, all three angles are 60°, and the area collapses to one clean formula in the side alone. That makes it a favorite in truss panels, paver patterns and anywhere a rigid, repeatable module is wanted. The formula looks like it fell from the sky, so the first job of this page is to show it did not.

Where the root three over four comes from

Drop the altitude from the apex to the base. It lands on the midpoint, splitting the triangle into two right triangles with hypotenuse s and short leg s/2. Pythagoras gives the height: h² = s² − (s/2)² = 3s²/4, so h = (√3/2)s, about 0.866 times the side.

Now use the ordinary area rule. A = ½ × base × height = ½ × s × (√3/2)s = (√3/4)s². The mysterious constant is just the half from the area rule multiplied by the root three over two from the height. Nothing new was invented, two familiar rules were composed.

1

Square

Square the side. For s = 10, s² = 100.

2

Multiply

Multiply by √3, which is 1.7321 to four places. That gives 173.21.

3

Divide

Divide by 4 to finish: 173.21 / 4 = 43.30 square units.

Two routes to the same area

The diagram works the side 10 triangle both ways. The shortcut lands on 43.30 directly from the side, and the long route through the 8.66 height lands on the same number. When a formula can be checked against its own derivation, use that as your error alarm.

An equilateral triangle with every side 10, its dashed altitude of 8.66 splitting the base into two segments of 5, beside the root three over four shortcut and half base times height both reaching 43.30.
The shortcut and the long way agree at 43.30. If your two routes ever disagree, the height step is the usual culprit.

Height is not the side

The tempting error is using the side as the height in half base times height. The slanted side is always longer than the vertical height, so the area comes out about 15 percent too big.

½ × 10 × 10givesside as height50.0

The 10 is a slant length. The true height is 8.66.

½ × 10 × 8.66givestrue height43.3

Height means perpendicular distance, never the slanted side.

A worked check you can repeat on paper

Take an equilateral paver with 300 mm sides and find its face area. Square the side: 300² = 90000 mm². Multiply by 1.7321: 155885. Divide by four: 38971 mm², call it 390 cm². A square meter of paving therefore needs roughly 1000000 / 38971 ≈ 25.7 pavers before waste.

Check by the long route. The height is 0.8660 × 300 = 259.8 mm, and ½ × 300 × 259.8 = 38971 mm². The two answers match to the last digit because they are algebraically the same calculation.

Scaling: why doubling the side quadruples the area

The side enters the formula squared, so area grows with the square of the side. Double the side and the area multiplies by four, triple it and the area multiplies by nine. This bites hardest in estimating: a gable that is half again as wide as the last one carries 2.25 times the area, not 1.5 times.

The same square law is why the formula is so sensitive to measurement error. A 2 percent error in the side becomes roughly a 4 percent error in the area. Measure the side well, because the formula will faithfully amplify whatever slop you feed it.

Mistakes that catch people out

  • Using the side as the height. The height is 0.866 times the side, always shorter. Substituting the side inflates the area by about 15 percent.
  • Forgetting to square. Multiplying the bare side by √3/4 gives a length, not an area, and it will be wildly small. Units are the tell: area needs squared units.
  • Halving twice. The half from the area rule is already inside the √3/4 constant. Applying another half on top silently cuts the answer in two.
  • Rounding √3 to 1.7. That alone introduces almost 2 percent of error. Use 1.7321 or the calculator key, and round the final answer instead.
  • Applying it to a triangle that is nearly equilateral. The formula is exact only when all three sides match. For a 10, 10, 11 triangle use Heron instead, and this shortcut only as a sanity band.

Try these

  1. Side 6. Find the area.
  2. Side 20. Find the area.
  3. Side 1. Find the area, the unit triangle.
  4. The side doubles from 5 to 10. By what factor does the area grow?
  5. Side 300 mm. Find the area in cm².
Show the answers
  1. (√3/4) × 36 = 15.588
  2. (√3/4) × 400 = 173.21
  3. √3/4 = 0.433
  4. four times, 10.825 to 43.30
  5. 389.7 cm²

Common questions

What is the formula for the area of an equilateral triangle?

Area equals root three over four times the side squared. With the side s, that is about 0.4330 times s squared, and it is exact, not an approximation.

Why does root three appear in the formula?

The altitude of an equilateral triangle is root three over two times the side, which comes straight from Pythagoras on half the triangle. Half base times that height produces the root three over four constant.

What is the height of an equilateral triangle?

Root three over two times the side, about 0.866 times the side. A side of 10 has a height of 8.660.

Can I use half base times height instead?

Yes, and you will get the identical answer, because the shortcut is half base times height with the height already expressed in terms of the side. It is a good independent check.

Does the formula work in any units?

Yes. Feed it meters and it returns square meters, feed it feet and it returns square feet. Just square the units along with the number.

What if my triangle is only approximately equilateral?

Use Heron on the three actual sides. The shortcut assumes perfect equality and drifts as the sides spread, so treat its answer as a check band rather than the result.

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