Area: Right-angled Triangle

Learn how a right-angled triangle forms half of a rectangle and practice calculating its area.

Right-angled Triangle

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

Learning Topics

📖 Area Study Guide

1. The Right-angled Triangle

Completing to a Rectangle & Congruence

αβαβabab
Completing to a Rectangle
If we take any right-angled triangle with legs a and b and copy it, we can flip the copy and place it next to the original to form a complete rectangle with side lengths a and b.
Definition: Congruent Triangles
Triangles are congruent when they are identical in shape and size, meaning all of their corresponding sides and angles match perfectly.
Since our original triangle and its flipped copy share these identical properties, these two triangles are congruent.

We can show this here using the Z-rule (alternate interior angles) we learned earlier:
• If the bottom angle of our original triangle is α\alpha and the top angle is β\beta (where α+β=90∘\alpha + \beta = 90^\circ), the Z-rule shows the opposite angles in the completed triangle must also be α\alpha and β\beta.
• Since both triangles together make a rectangle with all angles 90°, their opposite sides are equal, meaning the sides of both triangles are identical.

Because both triangles are congruent, they cover the exact same amount of space and have the same area. Therefore, the area of the right-angled triangle is exactly half of the rectangle's area:
A=a×b2=ab2A = \frac{a \times b}{2} = \frac{ab}{2}

2. Area Formulas

abA = ab2
Right-angled Triangle

Frequently Asked Questions

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.

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 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).

Area: Right-angled Triangle | SealMath