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Types of triangles: scalene, isosceles and equilateral

Scalene, isosceles and equilateral triangles — how to tell them apart, their angles, and how to find their area, including the area of an equilateral triangle.

2 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

A scalene triangle has three sides of different lengths — and therefore three different angles. It is one of three types by sides: scalene (no equal sides), isosceles (two equal sides) and equilateral (all three equal). Triangles are also named by angles: acute, right or obtuse.

The three types by sides

  • Scalene: all sides different, all angles different. Example: sides 5, 6, 7.
  • Isosceles: two equal sides, and the two angles opposite them are equal.
  • Equilateral: three equal sides and three 60° angles.
  • In every triangle the angles add up to 180°.

Worked example

Easy: area from base and height

Find the area of a triangle with base 10 cm and height 6 cm.

Show the working
  1. Use A=12bhA = \frac{1}{2}bh.
    12×10×6=30 cm2\tfrac{1}{2} \times 10 \times 6 = 30\text{ cm}^2

Answer: 30 cm230\text{ cm}^2

Checked by Sumzo’s math engine

Area of a scalene triangle from its three sides

When you know the three sides but not the height, use Heron’s formula: with s=a+b+c2s = \frac{a + b + c}{2} (half the perimeter), the area is s(s−a)(s−b)(s−c)\sqrt{s(s - a)(s - b)(s - c)}.

Worked example

Medium: a scalene triangle with sides 5, 6 and 7

Find the area of the scalene triangle with sides 5, 6 and 7, and describe its angles.

Show the working
  1. Half the perimeter.
    s=5+6+72=9s = \frac{5 + 6 + 7}{2} = 9
  2. Heron’s formula.
    A=9×4×3×2=216=66≈14.70A = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} = 6\sqrt{6} \approx 14.70
  3. The largest angle is opposite the 7: cos⁡C=52+62−722×5×6=0.2\cos C = \frac{5^2 + 6^2 - 7^2}{2 \times 5 \times 6} = 0.2, so C≈78.46°C \approx 78.46° — less than 90°, so the triangle is acute.

Answer: Area 66≈14.706\sqrt{6} \approx 14.70; a scalene, acute triangle.

Checked by Sumzo’s math engine

Area of an equilateral triangle

A=34s2A = \frac{\sqrt{3}}{4}s^2
Where $s$ is the length of each side.

Worked example

Hard: an equilateral triangle with side 6

Find the area of an equilateral triangle with side 6 cm.

Show the working
  1. The height splits it into two 30–60–90 triangles: h=62−32=27=33h = \sqrt{6^2 - 3^2} = \sqrt{27} = 3\sqrt{3}.
  2. Area.
    12×6×33=93≈15.59 cm2\tfrac{1}{2} \times 6 \times 3\sqrt{3} = 9\sqrt{3} \approx 15.59\text{ cm}^2
  3. Same with the formula: 34×36=93\frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} ✓.

Answer: 93≈15.59 cm29\sqrt{3} \approx 15.59\text{ cm}^2

Checked by Sumzo’s math engine

Common mistakes

  • Calling a right triangle “not scalene” — a 3–4–5 triangle is both right and scalene.
  • Using a slanted side as the height.
  • Forgetting to halve the perimeter in Heron’s formula.
  • Assuming an isosceles triangle’s equal angles are next to the equal sides — they are opposite them.

Frequently asked questions

What is a scalene triangle?

A triangle whose three sides all have different lengths, so its three angles are all different too.

Can a scalene triangle have a right angle?

Yes. The 3–4–5 triangle is a scalene right triangle: all sides differ and one angle is 90°.

What is the area of an equilateral triangle?

34s2\frac{\sqrt{3}}{4}s^2. With side 6, that is 93≈15.599\sqrt{3} \approx 15.59.

Calculators

Practice

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