Problem 7 - Basic Engineering Mathematics

Question

Find the areas of the circles having (a) a radius of \(5 \mathrm{~cm}\), (b) a diameter of \(15 \mathrm{~mm}\), (c) a circumference of \(70 \mathrm{~mm}\).

Step-by-Step Solution

Answer

(a) \( 25\pi \ \mathrm{cm}^2 \), (b) \( 56.25\pi \ \mathrm{mm}^2 \), (c) \( \frac{1225}{\pi} \ \mathrm{mm}^2 \).

Step 1: Formula for Area of a Circle

The area of a circle is given by the formula \( A = \pi r^2 \), where \( A \) is the area, and \( r \) is the radius.

Step 2: Problem (a) - Area with Known Radius

For part (a), we are given the radius \( r = 5 \) cm. Substitute this value into the formula: \( A = \pi (5)^2 = 25\pi \). Thus, the area is \( 25\pi \ \mathrm{cm}^2 \).

Step 3: Problem (b) - Convert Diameter to Radius

For part (b), we are given the diameter \( d = 15 \) mm. Since radius is half of diameter, \( r = \frac{d}{2} = \frac{15}{2} = 7.5 \) mm.

Step 4: Problem (b) - Area with Calculated Radius

Now use the radius to find the area: \( A = \pi (7.5)^2 = 56.25\pi \). Thus, the area is \( 56.25\pi \ \mathrm{mm}^2 \).

Step 5: Problem (c) - Determine Radius from Circumference

For part (c), we are given the circumference \( C = 70 \) mm. The formula for circumference is \( C = 2\pi r \). Solve for the radius \( r \): \( r = \frac{C}{2\pi} = \frac{70}{2\pi} = \frac{35}{\pi} \).

Step 6: Problem (c) - Area with Calculated Radius

Substitute the radius back into the area formula: \( A = \pi \left(\frac{35}{\pi}\right)^2 = \frac{1225}{\pi} \). Thus, the area is \( \frac{1225}{\pi} \ \mathrm{mm}^2 \).

Key Concepts

Mathematical Formula

Understanding mathematical formulas is crucial for solving geometry problems like finding the area of a circle. The formula for the area of a circle is \[A = \pi r^2\]where:

• \(A\) stands for the area of the circle.
• \(\pi\) is a constant approximately equal to 3.14159.
• \(r\) represents the radius of the circle.

The significance of this formula lies in its use to calculate the space inside the circular boundary. This mathematical concept is essential for many real-world applications, from designing round tables to allocating land parcels. Recognizing the formula and knowing how to apply it can simplify and solve problems involving circular shapes.

Circle Geometry

Circle geometry is a fascinating area of mathematics focusing on the properties and measures of circles. A circle is a two-dimensional shape defined as the set of all points equidistant from a fixed point called the center. Key elements in circle geometry include:

• Radius (\(r\)): The distance from the center of the circle to any point on its circumference.
• Diameter (\(d\)): The longest distance across the circle, passing through the center. It is twice the radius \((d = 2r)\).
• Circumference (\(C\)): The total distance around the circle, calculated by the formula \(C = 2\pi r\).

These elements are interconnected, allowing us to compute one quantity knowing another. For instance, if you know a circle's diameter, you can easily find its radius and use it in other formulas.

Radius and Diameter Calculations

Calculating the radius and diameter of a circle is foundational to solving many circle-related problems. The radius is half the diameter, which can be expressed by the equation \[r = \frac{d}{2}\]where \(d\) is the diameter. This relationship underscores the symmetry and simple proportionality prevalent in circular shapes.
To perform calculations:- **When given the diameter**, divide by 2 to find the radius as shown: \(r = \frac{15}{2} = 7.5 \) mm.- **When given the circumference**, rearrange the formula for circumference (\(C = 2\pi r\)) to solve for the radius: \(r = \frac{C}{2\pi}\). For example, with a circumference of 70 mm: \(r = \frac{70}{2\pi} \approx 11.14 \) mm.
Understanding these relationships ensures efficient and accurate problem-solving in scenarios involving circles.