Question

The area \(A\) of a triangle whose base is \(b\) and whose altitude is \(h\) is \(A=\) _____.

Short answer

The area of the triangle is \( A = \frac{1}{2} \times b \times h \)

Step-by-step solution

Identify the Formula

To find the area of a triangle, use the formula: \[ A = \frac{1}{2} \times b \times h \] where \(A\) is the area, \(b\) is the base, and \(h\) is the height (altitude) of the triangle.

Apply the Variables

Substitute the given values of base (\(b\)) and height (\(h\)) into the formula. These values will be provided in the problem or given as specific numbers.

Calculate the Area

Multiply the base \(b\) by the height \(h\), and then divide the result by 2 to find the area: \[ A = \frac{1}{2} \times b \times h \]

Key concepts

Triangle area formula

To find the area of a triangle, you can use a very straightforward formula. The formula is: \( A = \frac{1}{2} \times b \times h \). This formula involves three key components:

• A: Area of the triangle
• b: Length of the base
• h: Height (also called altitude)

You simply multiply the base length by the height, and then divide the result by 2.
This formula is particularly useful because it applies to all types of triangles, whether they are scalene, isosceles, or equilateral.
For instance, if the base of a triangle is 8 units long and the height is 5 units, you would calculate the area like this:

\( A = \frac{1}{2} \times 8 \times 5 = 20 \).

Understanding this formula allows you to quickly determine how much space the triangle occupies.

Base and height

In the triangle area formula, base and height are crucial elements. They ensure that you're calculating the area correctly.
The base of the triangle is any one of its sides. Generally, we choose the side that makes measuring the height easier.
The height (or altitude) is a perpendicular line from the base to the opposite vertex of the triangle.

• The base can be any length, just as height ensures to form a right angle with the base.
• Imagine the base as the 'bottom' edge and the height as a 'vertical' line going up to point of the triangle.

This property guarantees that your area calculation is based on the largest shape that can fit within the triangle.
If you have a triangle with a base of 10 units and a height of 7 units, using them with the formula will give you the area.

Never forget: Both the base and the height must be measurements in the same units. This will ensure your area is accurately calculated.

Multiplication and division in geometry

Understanding multiplication and division in geometry is essential to mastering formulas like that for the area of a triangle.These arithmetic operations allow us to handle spatial dimensions with ease.

• Multiplication helps us combine two measurements (base and height) to find a proportion of the entire shape.
• Division, specifically by 2 in our triangle formula, adjusts the final value to represent a half-shape since a triangle is essentially half of a rectangle.

When you multiply the base by the height in the formula \( \frac{1}{2} \times b \times h \), it means you're calculating what the equivalent rectangle would be.
Then, dividing by 2 scales it back to match the triangle.
This method efficiently turns complex shapes into manageable calculations, demonstrating the elegance of mathematical operations.

Always take note: These operations must be performed step-by-step (multiply first, then divide), ensuring no part of the calculation is skipped.