Question

Find the area \(A\) of a triangle with height 14 inches and base 4 inches.

Short answer

The area is 28 square inches.

Step-by-step solution

- Identify the Formula

To find the area of a triangle, use the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

- Substitute the Values

Substitute the given height and base into the formula. Here, the base is 4 inches and the height is 14 inches. So, it becomes: \[ A = \frac{1}{2} \times 4 \text{ inches} \times 14 \text{ inches} \]

- Perform the Multiplication

First, multiply the base and height: \[ 4 \text{ inches} \times 14 \text{ inches} = 56 \text{ square inches} \]

- Divide by 2

Now, divide the product by 2 to find the area of the triangle: \[ A = \frac{56 \text{ square inches}}{2} = 28 \text{ square inches} \]

- State the Answer

The area of the triangle is 28 square inches.

Key concepts

Triangle Area Formula

To calculate the area of a triangle, you can use a straightforward formula. This formula is derived from the basic geometric principles of shapes. The formula is written as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Let's break it down to make it easier to understand:

• \(A\): Represents the area of the triangle.
• Base: The length of the bottom side of the triangle.
• Height: The perpendicular distance from the base to the topmost point of the triangle.

The fraction \(\frac{1}{2}\) is crucial because a triangle is essentially half of a rectangle when the height is perpendicular to the base. This is why the area formula includes this fraction.
Understanding this formula allows you to quickly find the area of any triangle if you know the base and height.

Height and Base in Triangles

In the context of triangles, the terms 'height' and 'base' are essential. Grasping their meanings simplifies your calculations:

• Base: Choose any side of the triangle as a base. For consistency, it's often the lowest side.
• Height: This is always perpendicular to the base. Imagine drawing a line from the opposite vertex (top point) straight down to form a right angle with the base.

For example, if you have a triangle with a base of 4 inches and a height of 14 inches, it may look like this:

• The 'base' is the flat bottom part, 4 inches long.
• The 'height' is a straight line from the top vertex down to the base, measuring 14 inches.

Remember, the choice of base is flexible, but the corresponding height must always be perpendicular to it.

Multiplication and Division in Geometry

Understanding multiplication and division is critical in geometry, particularly when using formulas like the one for the area of a triangle. Let's see how they apply:

• Multiplication: When finding the area, you'll first multiply the base by the height. In our exercise, this was: \(4 \text{ inches} \times 14 \text{ inches} = 56 \text{ square inches}\).
• Division: After multiplying, you must divide the product by 2. This step reduces the rectangle's area (if you imagine doubling the triangle to form a rectangle) to just the triangle's area. In the example, it became: \(\frac{56 \text{ square inches}}{2} = 28 \text{ square inches}\).

The use of these operations ensures you correctly apply the triangle area formula. Always perform multiplication first, then proceed with division (following basic order of operations rules).
By mastering these steps, solving for the area of any triangle becomes a straightforward task.