Question

Bianca calculated the height of the equilateral triangle with side lengths of 10.

tangent (30) = StartFraction 5 Over h EndFraction An equilateral triangle with side lengths of 10 is shown. A bisector is drawn to split the side into 2 equal parts and splits the angle into 2 30 degree segments.
Then, she used the formula for area of a triangle to approximate its area, as shown below.

A = one-half b h. = one-half (10) (8.7). = 43.5 units squared.

Calculate the area of the equilateral triangle using the formula for area of a regular polygon, and compare it to Bianca’s answer.

The apothem, rounded to the nearest tenth, is
✔ 2.9
units.

The perimeter of the equilateral triangle is
✔ 30
units.

Therefore, the area of the equilateral triangle is
, or approximately 43.5 units2.

The calculated areas are
✔ the same, despite using different formulas

Answer 1

Use A = 1/2 · apothem · perimeter.

For an equilateral triangle of side s = 10:
- apothem a = s/(2 tan(π/3)) = 10/(2·√3) = 5/√3 ≈ 2.8868 (≈ 2.9 rounded).
- perimeter P = 3·10 = 30.

So
A = 1/2 · a · P = 1/2 · (5/√3) · 30 = 75/√3 = 25√3 ≈ 43.3013 units^2.

Comparison with Bianca:
- Bianca’s area 43.5 arises from rounding the height/apothem (h ≈ 8.6603 rounded to 8.7, or a ≈ 2.8868 rounded to 2.9).
- The exact area is 25√3 ≈ 43.30; Bianca’s 43.5 is about 0.20 units^2 (≈0.46%) larger.
- So they agree to one decimal place (both ≈ 43.3 or 43.5 depending on rounding), but are not exactly the same.