Area: Special Triangles

Learn about isosceles, equilateral, and 30-60-90 triangles. Practice finding their area, height, or sides using the HL / RHS congruence rule and Pythagorean theorem.

Special Triangles

Use the workspace below. Write equations like A = 30 to solve for the area.

Learning Topics

📖 Area Study Guide

1. Right-Angle Congruence Rule: HL / RHS

Two right-angled triangles are congruent (identical in size and shape) if they satisfy the HL or RHS rule:

• HL (Hypotenuse-Leg):
- H (Hypotenuse): The hypotenuses are equal in length.
- L (Leg): One of the other sides (legs) is equal in length.

• RHS (Right angle-Hypotenuse-Side):
- R (Right angle): Both triangles have a

90^\circ

angle.
- H (Hypotenuse): The hypotenuses are equal in length.
- S (Side): One of the other sides (legs) is equal in length.

This allows us to prove properties of other triangles by splitting them into two right-angled halves.

2. Isosceles Triangle (Altitude Bisection)

An isosceles triangle is a triangle with at least two equal legs of length

b

. In geometric diagrams, equal sides are often indicated by matching tick marks (small slash tags) drawn on them. For example, if you see a single slash mark on two sides, it indicates that their lengths are identical.

If we draw the altitude (height

h

) from the apex perpendicular to the base

a

, it splits the triangle into two right-angled triangles:
• Both halves share the altitude

h

as a common leg.
• Both halves have equal hypotenuses (the equal legs of length

b

).

By the HL / RHS rule, these two halves are congruent! This means the altitude splits the base

a

into two equal halves of length

\frac{a}{2}

Altitude of Isosceles Triangle Formula

Using the Pythagorean theorem on one half:

\left(\frac{a}{2}\right)^2 + h^2 = b^2

From this, we can derive the height

h

if we know base

a

and leg

b

, or find the base

a

a = 2\sqrt{b^2 - h^2}

Altitude of an Isosceles Triangle

h = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}

3. Equilateral Triangle (Special Case of Isosceles)

An equilateral triangle is a special case of an isosceles triangle where all three sides are equal to length

a

.

Since it is isosceles, we can draw the height

h

from the apex, splitting the base into two equal halves of length

\frac{a}{2}

. The hypotenuse is the side length

a

Equilateral Triangle Law

1. If a triangle is equilateral, all its angles are

60^\circ

.
2. Conversely, if all angles of a triangle are

60^\circ

, the triangle is equilateral.

Height of Equilateral Triangle Formula

By applying the Pythagorean theorem to one half:

\left(\frac{a}{2}\right)^2 + h^2 = a^2 \implies \frac{a^2}{4} + h^2 = a^2

h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}

Height of an Equilateral Triangle

h = a \frac{\sqrt{3}}{2}

Area of Equilateral Triangle Formula

We can now calculate the area of the equilateral triangle using the base

a

and derived height

h

\text{Area} = \frac{b \times h}{2} = \frac{a \times \left(a\dfrac{\sqrt{3}}{2}\right)}{2}

Area of an Equilateral Triangle

A = a^2 \frac{\sqrt{3}}{4}

4. The 30-60-90 Right Triangle

If we cut the equilateral triangle (with side

c

) in half using the altitude, we obtain a right-angled triangle with angles $30^\circ$ , $60^\circ$ and $90^\circ$ .

In this triangle, the hypotenuse is the original side length

c

, and the shortest leg

a

is exactly half of

c

(as established during our bisection derivation above).

30-60-90 Triangle Law

In any 30-60-90 triangle, the leg opposite the

30^\circ

angle (the shortest leg) is always half the length of the hypotenuse:

a = \frac{c}{2}

(or

c = 2a

30-60-90 Triangle Area Formula

By Pythagoras, the longer leg (opposite

60^\circ

) is

b = a\sqrt{3}

. The area is:

\text{Area} = \frac{a \times (a\sqrt{3})}{2}

Area of a 30-60-90 Triangle

A = a^2 \frac{\sqrt{3}}{2}

where a is the shortest leg.

Note on 4th Roots

When solving certain problems (such as finding the height of an isosceles triangle from its area and the leg-to-height ratio), you might encounter equations of the form $h^4 = x$ . To solve for $h$ , you must take the fourth root of both sides: $h = \sqrt[4]{x}$ . For example, if $h^4 = 625$ , then $h = \sqrt[4]{625} = 5$ .

Mastering SealMath: Entering Custom Roots

To enter a third root, fourth root, or any other nth root, you have several options:

Keyboard shortcut: Type root or nthroot in the input box. MathLive will instantly create the root symbol $\sqrt[\scriptstyle\square]{\square}$ with the cursor inside the index box — type the root index (e.g., 4), then press the right arrow key to move inside the root and type your number.
Virtual keyboard: Click the ⌨️ keyboard icon inside the input box to open the on-screen keyboard, then press the $\sqrt[\scriptstyle y]{x}$ button found under the math/symbols tab.
In the Scientific Calculator: Use the nth root function nrt(index, value). For example, to calculate the fourth root of 16, type nrt(4, 16). Alternatively, use fractional exponents: 16^(1/4). You can also copy and paste LaTeX like \sqrt[4]{16} directly into the calculator. There is also a dedicated button: press Shift, then find the third button from the right on the second row of the calculator.

Frequently Asked Questions

What are the main types of special triangles?

The three main types are: isosceles (two equal sides and two equal base angles), equilateral (all sides and angles equal — each angle is 60°), and the 30-60-90 right triangle (fixed side ratios of 1 : √3 : 2).

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). It is used to find a missing side length when the other two are known.