The area of any triangle is ½ × base × height, where the height is the perpendicular distance from the chosen base to the opposite vertex — not a slanted side. In a right triangle the two legs are perpendicular, so they serve directly as base and height and the area is simply ½ × leg × leg. For an equilateral triangle of side s, the height is (s√3)/2, which gives the tidy formula Area = (√3/4) × s². A frequent GRE twist runs the formula backward: given the area and one dimension, solve for the other. Watch the on-screen basic calculator here — halving and multiplying are quick by hand, so reserve the calculator for the genuinely awkward numbers.
✅ Solved examples
1. A triangle has a base of 10 and a height of 6. Find its area.
Area = ½ × base × height = ½ × 10 × 6 = 30.
2. A right triangle has legs of 5 and 12. Find its area.
The legs are perpendicular, so Area = ½ × 5 × 12 = 30.
3. Find the area of an equilateral triangle with a side of 4.
Area = (√3/4) × s² = (√3/4) × 16 = 4√3.
4. A triangle has an area of 20 and a base of 8. Find its height.
Area = ½ × base × height → 20 = ½ × 8 × h → 20 = 4h → h = 5.
✏️ Practice — try these, take hints as needed
1. A triangle has a base of 14 and a height of 5. Find its area.
Area = ½ × base × height.
½ × 14 × 5.
7 × 5.
35
2. A right triangle has legs of 6 and 8. Find its area.
The legs are the base and height.
½ × 6 × 8.
3 × 8.
24
3. Find the area of an equilateral triangle with a side of 6.
Area = (√3/4) × s².
s² = 36.
(√3/4) × 36 = 9√3.
9√3
4. A triangle has an area of 54 and a base of 12. Find its height.
54 = ½ × 12 × h.
54 = 6h.
Divide by 6.
9
5. A triangle has an area of 30 and a height of 5. Find its base.
30 = ½ × base × 5.
30 = 2.5 × base.
Divide by 2.5.
12
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Vidaara uses essential cookies to run the site and, with your consent, optional cookies to understand how learners use Vidaara so we can improve it. We never sell your data. Read our Cookie Policy and Privacy Policy.