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∘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)

bba2a2h
An isosceles triangle is a triangle with at least two equal legs of length bb.

If we draw the altitude (height hh) from the apex perpendicular to the base aa, it splits the triangle into two right-angled triangles:
• Both halves share the altitude hh as a common leg.
• Both halves have equal hypotenuses (the equal legs of length bb).

By the HL / RHS rule, these two halves are congruent! This means the altitude splits the base aa into two equal halves of length a2\frac{a}{2}.
Altitude of Isosceles Triangle Formula
Using the Pythagorean theorem on one half:
(a2)2+h2=b2\left(\frac{a}{2}\right)^2 + h^2 = b^2

From this, we can derive the height hh if we know base aa and leg bb:
Altitude of Isosceles Triangle:
h=b2−(a2)2\boxed{h = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}}
or find the base aa: a=2b2−h2a = 2\sqrt{b^2 - h^2}.

3. Equilateral Triangle (Special Case of Isosceles)

aaah60°
An equilateral triangle is a special case of an isosceles triangle where all three sides are equal to length aa (and all angles are 60∘60^\circ).

Since it is isosceles, we can draw the height hh from the apex, splitting the base into two equal halves of length a2\frac{a}{2}. The hypotenuse is the side length aa.
Height of Equilateral Triangle Formula
By applying the Pythagorean theorem to one half:
(a2)2+h2=a2  ⟹  a24+h2=a2\left(\frac{a}{2}\right)^2 + h^2 = a^2 \implies \frac{a^2}{4} + h^2 = a^2

h2=a2−a24=3a24h^2 = a^2 - \frac{a^2}{4} = \frac{3a^2}{4}

Height of Equilateral Triangle:
h=a32\boxed{h = a\frac{\sqrt{3}}{2}}
Area of Equilateral Triangle Formula
We can now calculate the area of the equilateral triangle using the base aa and derived height hh:
Area=b×h2=a×(a32)2\text{Area} = \frac{b \times h}{2} = \frac{a \times \left(a\dfrac{\sqrt{3}}{2}\right)}{2}

Area of Equilateral Triangle:
Area=a234\boxed{\text{Area} = a^2\frac{\sqrt{3}}{4}}

4. The 30-60-90 Right Triangle

b = a√3ac = 2a60°30°
If we cut the equilateral triangle (with side cc) in half using the altitude, we obtain a right-angled triangle with angles 30∘30^\circ, 60∘60^\circ and 90∘90^\circ.

In this triangle:
• The hypotenuse is the original side length cc.
• The shortest leg (opposite the 30∘30^\circ angle) is exactly half the base of the equilateral triangle, which is c2\frac{c}{2}. Therefore, in any 30-60-90 triangle, the leg opposite the 30∘30^\circ angle is always half the length of the hypotenuse: a=c2a = \frac{c}{2} (or c=2ac = 2a).
30-60-90 Triangle Area Formula
By Pythagoras, the longer leg (opposite 60∘60^\circ) is b=a3b = a\sqrt{3}. The area is:
Area=a×(a3)2\text{Area} = \frac{a \times (a\sqrt{3})}{2}

30-60-90 Triangle Area:
Area=a232\boxed{\text{Area} = a^2\frac{\sqrt{3}}{2}}
where aa 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 h4=xh^4 = x. To solve for hh, you must take the fourth root of both sides: h=x4h = \sqrt[4]{x}. For example, if h4=625h^4 = 625, then h=6254=5h = \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 xy\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

How is the area of a shape defined?

The area of a shape is defined by how many unit squares of 1 by 1 fit inside it. For example, if a rectangle can be divided exactly into 30 squares of 1 by 1, its area is 30.

How do you calculate the area of a rectangle and a square?

The area of a rectangle is calculated as width × height (A = w × h). A square is a special type of rectangle where all sides are equal (w = h = s). Thus, the area of a square is side × side, or side squared (A = s²).

How do you calculate the area of a right-angled triangle?

The area of a right-angled triangle is calculated by multiplying its two perpendicular legs and dividing by 2 (A = ab / 2). This is because a right-angled triangle is exactly half of a rectangle with the same width and height.

How do you calculate the area of a general triangle?

The area of any general triangle is calculated as half the base times the height (A = (1/2) × b × h or A = bh/2), where the height is the perpendicular distance from the base to the opposite vertex.

How do you calculate the area of a circle?

The area of a circle is calculated as π times the radius squared (A = πr²). Since π (pi) is an irrational number, the area of a circle with a rational radius will always be an irrational number.

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.

What is the area ratio of inscribed circles and squares?

For a square inscribed in a circle, the area ratio of the square to the circle is always exactly 2/π ≈ 0.637. For a circle inscribed in a square, the area ratio of the circle to the square is always exactly π/4 ≈ 0.785. These ratios are constant regardless of the shapes' actual sizes.

Can the area of a shape be an irrational number?

Yes, if the side lengths are irrational numbers (such as √2), the resulting area can be either rational or irrational. You can learn more about these classifications in our Number Sets - Real & Complex topic.

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).

Special Triangles | SealMath | SealMath