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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

A right-angled triangle (or right triangle) is a triangle where one of the angles is exactly 90° (a right angle). The two sides that meet at the right angle are called its perpendicular legs (which we can label as a and b).

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 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.

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.

Area: Right-angled Triangle | SealMath