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.
📖 Area Study Guide
1. Base and Height of a Triangle
• 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.
In an acute triangle (where all angles are less than 90°), the height to the base falls inside the triangle.
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)
3. Triangle Area Formula
Where:
• is the length of the base (opposite side).
• 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.
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).
Why does 1 cm² equal 100 mm² and not just 10 mm²?
Because area is two-dimensional (length × length), the conversion factor must be applied twice. Since 1 cm = 10 mm, we have 1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm². In general, if 1 unit = k sub-units, then 1 unit² = k² sub-units². The same logic applies to all other area unit conversions.