Question

Find the area of each triangle. Round answers to two decimal places. $$a=4, \quad b=4, \quad c=4$$

Short answer

The area of the triangle is approximately 6.93 square units.

Step-by-step solution

- Identify the formula for the area of a triangle using Heron's formula

Heron's formula is used to find the area of a triangle when the lengths of all three sides are known. The formula is: \[ A = \sqrt{s(s-a)(s-b)(s-c)} \] where \( s \) is the semi-perimeter of the triangle.

- Calculate the semi-perimeter

The semi-perimeter \( s \) is calculated as: \[ s = \frac{a + b + c}{2} \] Given \( a = 4 \), \( b = 4 \), \( c = 4 \), we have: \[ s = \frac{4 + 4 + 4}{2} = 6 \]

- Substitute values into Heron's formula

Using the values of \( s \) and the sides \( a, b, c \), we substitute into Heron's formula: \[ A = \sqrt{6(6-4)(6-4)(6-4)} = \sqrt{6 \cdot 2 \cdot 2 \cdot 2} = \sqrt{48} \]

- Compute the square root and round to two decimal places

The area \( A \) is calculated as: \[ A = \sqrt{48} \approx 6.93 \] Rounded to two decimal places, the area of the triangle is approximately 6.93.

Key concepts

Area of a Triangle

To find the area of a triangle when you know the lengths of all three sides, you can use Heron's formula. This formula is very handy and works for any triangle, whether it's right-angled, obtuse, or acute. The general form of Heron's formula is:

\( A = \sqrt{s(s-a)(s-b)(s-c)} \)

where \( A \) is the area and \( s \) is the semi-perimeter of the triangle. First, you calculate the semi-perimeter, then plug all known values into the formula to find the area. This method breaks down a complex process into easy, manageable steps, making it possible to compute the area using basic arithmetic and square root operations.

For example, if each side of the triangle is 4 units (an equilateral triangle), you would first determine the semi-perimeter and then use it to find the area as detailed in the exercise solution.

Semi-Perimeter Calculation

The semi-perimeter \( s \) is half of the triangle's perimeter. It's a crucial value needed in Heron's formula for calculating the area. The formula for the semi-perimeter is given by:

\[ s = \frac{a + b + c}{2} \]

Here, \( a, b, \) and \( c \) are the lengths of the sides of the triangle. Breaking down the calculation:

• Add up the lengths of all three sides.
• Divide the sum by 2 to get the semi-perimeter.

Using the given example:

\ a = 4, b = 4, c = 4\
\[ s = \frac{4 + 4 + 4}{2} = 6 \]

With this semi-perimeter value, you can move on to find the area of the triangle using Heron's formula. This concept simplifies the initial steps toward calculating the area, transforming a seemingly tough geometry problem into basic arithmetic.

Square Root

The square root function is used in Heron's formula to finalize the area calculation. After finding the semi-perimeter and substituting all values into the Heron’s formula, the process involves calculating the inner product and then taking the square root of the result. Let's break it down further:

When using Heron's formula \( A = \sqrt{s(s-a)(s-b)(s-c)} \), you arrive at:

\[ A = \sqrt{6(6-4)(6-4)(6-4)} = \sqrt{6 \cdot 2 \cdot 2 \cdot 2} = \sqrt{48} \]

Finally, to find \( \sqrt{48} \), use a calculator or estimate by finding the nearest perfect squares. You obtain:

\[ \sqrt{48} \approx 6.93 \]

This result is the area of the triangle, rounded to two decimal places. Understanding how to handle the square root operation is key, as it transforms the summed products into a manageable result, giving you the precise area you need. Whether you use a calculator or do it manually, this concept ensures you can complete the problem accurately.