Question

The formula for the area of a triangle is \(1 / 2 \times\) base \(\times\) height. What is the area of a triangle in square meters if its base is \(166 \mathrm{~mm}\) and its height is \(930.0 \mathrm{~mm}\) ? Express the answer to the proper number of significant figures.

Short answer

The area is 0.0772 square meters.

Step-by-step solution

Understanding the Problem

We are asked to find the area of a triangle given its base and height in millimeters. The formula for the area is \( \frac{1}{2} \times \text{base} \times \text{height} \). The final area should be converted to square meters.

Convert Measurements

Convert the base and height from millimeters (mm) to meters (m) since 1 meter is 1000 millimeters. Thus, base in meters is \(0.166 \) m and height in meters is \(0.930 \) m.

Use the Area Formula

Substitute the converted values into the area formula: \[\text{Area} = \frac{1}{2} \times 0.166 \times 0.930.\]

Calculate the Area

Now compute the area: \[\text{Area} = \frac{1}{2} \times 0.166 \times 0.930 = 0.07719 \text{ square meters}.\]

Apply Significant Figures

The base has 3 significant figures (0.166), and the height has 4 significant figures (0.9300). Our final answer should be expressed using the lower number of significant figures, which is 3. Thus, the area is \(0.0772 \) square meters.

Key concepts

Understanding Significant Figures

Significant figures are an important concept in mathematics and science, as they indicate the precision of a measurement. Significant figures reflect meaningful digits in a number and help to convey the accuracy of measurements during calculations.

In any given measurement, significant figures include all the digits that are known with certainty, plus one final digit that is estimated. When performing calculations, it is crucial to ensure that the result is expressed with the correct number of significant figures. This is because the precision of a result can only be as accurate as the least precise measurement.

For instance, in our problem, we have two measurements: the base of the triangle is 0.166 m (3 significant figures) and the height is 0.930 m (4 significant figures). The result for the area calculation must be expressed in the lower number—3 significant figures—because that is the precision of the least precise measurement. This means even if our calculated area is 0.07719 square meters, we round it to 0.0772 square meters to reflect 3 significant figures. This helps us maintain a consistent level of precision throughout the calculation.

The Importance of Unit Conversion

Unit conversion is a foundational skill in mathematics and science, as it allows us to work with measurements in different units. Converting units is critical for ensuring that all parts of a calculation are consistent, which often involves expressing measurements in the correct units before performing any operations.

In our exercise, the triangle's base and height were initially given in millimeters, a subunit of meters, the standard unit for measuring length in the metric system. Since there are 1000 millimeters in one meter, we need to divide the original measurements by 1000 to convert from millimeters to meters.

• Base: 166 mm = 0.166 m
• Height: 930 mm = 0.930 m

Converting these units is vital for using the standard triangle area formula (\(1/2 \times \text{base} \times \text{height}\)) effectively. Without this conversion, our calculated area would not be expressed in the correct square meter unit, leading to potentially inaccurate interpretations of size.

Applying Geometry to Solve Problems

Geometry, a branch of mathematics concerning shapes, sizes, and properties of space, provides us with structured ways to calculate areas, perimeters, and volumes. In this problem, we focus on finding the area of a triangle using its base and height, demonstrating practical applications of geometric principles.

The area of a triangle is calculated using the formula: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). This formula comes from the understanding of a triangle as half of a rectangle when the triangle is right-angled. The base and height serve as two perpendicular sides, similar to length and width in a rectangle.

Triangles can take various shapes, including equilateral, isosceles, and scalene. Regardless of type, the base and height are sufficient to apply the area formula. It's essential to note that the height must be perpendicular to the base to yield a valid area result. This attentiveness to detail in geometry ensures accuracy in our calculations and understanding of spatial dimensions.