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Area: Triangle

Learn how to calculate the area of any triangle using the base and height, and practice dynamic exercises.

Triangle

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

Learning Topics

📖 Area Study Guide

1. Base and Height of a Triangle

To calculate the area of any triangle, we need two key measurements:
• Base (b): Any of the triangle's three sides.
• Height (h): The perpendicular line segment from the opposite vertex to that base (or its extension). The height is always at a 90° angle to the base.
bh
Acute Triangle (Height Inside)

In an acute triangle (where all angles are less than 90°), the height to the base falls inside the triangle.

bh
Obtuse Triangle (Height Outside)

In an obtuse triangle (where one angle is greater than 90°), the height to the base can fall outside the triangle. To measure it, we extend the base line outwards.

2. Intersection of Heights (Orthocenter)

Every triangle has three heights (one for each of its three sides). An elegant property of geometry is that all three heights (or their extensions) intersect at exactly one point. This intersection point is called the orthocenter.
ABCOrthocenter

3. Triangle Area Formula

Since any triangle is exactly half of a parallelogram (or rectangle) with the same base and height, its area is given by multiplying the base and height and dividing by 2:
A=b×h2A = \frac{b \times h}{2}

Where:
• bb is the length of the base (opposite side).
• hh is the height perpendicular to that base.

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: Triangle | SealMath