Let's solve this geometry problem step-by-step to find the circumference of the circle.

Understanding Circle Properties

A circle has several key properties:

• Radius (r): The distance from the center of the circle to any point on its edge.
• Diameter (d): The distance across the circle passing through the center. It is equal to twice the radius.
• Circumference (C): The distance around the edge of the circle.

The relationship between these properties is important:

• Diameter = 2 × Radius (d = 2r)
• Circumference = $\pi$ × Diameter (C = $\pi$d)
• Circumference = 2 × $\pi$ × Radius (C = 2$\pi$r)

We typically use the value of $\pi$ as approximately 22/7 or 3.14.

Setting Up the Problem: Sum of Radius and Diameter

The question gives us that the sum of the radius (r) and the diameter (d) of the circle is 84 cm. We can write this as an equation:

r + d = 84 cm

Solving for the Radius

We know that the diameter is twice the radius (d = 2r). We can substitute this into the equation:

r + (2r) = 84

Combine the terms involving 'r':

3r = 84

Now, solve for the radius 'r' by dividing both sides by 3:

r = $\frac{84}{3}$

r = 28 cm

So, the radius of the circle is 28 cm.

Calculating the Circumference

Now that we have the radius (r = 28 cm), we can find the circumference (C) using the formula C = 2$\pi$r. We will use the common approximation for $\pi$, which is 22/7.

C = 2 × $\frac{22}{7}$ × 28

We can simplify the calculation by dividing 28 by 7:

C = 2 × $\frac{22}{1}$ × $\frac{28}{7}$

C = 2 × 22 × 4

Multiply the numbers together:

C = 44 × 4

C = 176 cm

The circumference of the circle is 176 cm.

Summary of Calculations

The calculated circumference is 176 cm. Let's compare this with the given options.

• Option 1: 84 cm
• Option 2: 176 cm
• Option 3: 168 cm
• Option 4: 88 cm

The calculated value matches Option 2.