Question

A piece of wire 32 cm long is bent to form the figure given below. APD is a semicircle and AB = BC = CD. Find the radius and area of the figure.

Answer 1

Step 1: Find the value of x

Let's call the length of the straight sides x. The problem says that the three straight sides are equal: AB = BC = CD = x.

The curved part is a semicircle. Its diameter is the line connecting A and D. From the picture, we can see that the length of this diameter is the same as the length of the bottom side, BC. 

So, the diameter of the semicircle is also x.

The total length of the wire is 32 cm. This length is made up of the three straight sides and the curved semicircle.

The length of the three straight sides is x + x + x = 3x.

The length of the semicircle is half the circumference of a circle:

`1/2 xx pi  xx "diameter" = 1/2 xx pi  xx = (pix)/2`

Now we can set up an equation for the total length of the wire:

`3x + (pix)/2 = 32`

To solve for x, let's use the value of π ≈ 3.14

`3x + (3.14x)/2 = 32`

3x + 1.57x = 32

4.57x = 32

`x = 32/4.57 ≈ 7` cm

Step 2: Find the radius

The radius of the semicircle is half of its diameter.

• Diameter = x = 7 cm
• Radius = `x/2 = 7/2 = 3.5` cm

Step 3: Find the area of the figure

The total area of the figure is the sum of the area of the square (the bottom part) and the area of the semicircle (the top part).

Area of the square: The sides of the square are all equal to x.

Area = x² = 7 × 7 = 49 cm²

Area of the semicircle: The formula is `1/2 xx pi xx "radius"^2`.

`"Area" = 1/2 xx 22/7 xx (3.5)^2`

`"Area" = 1/2 xx 22/7 xx 12.25 = 19.25  cm^2`

Total Area = Area of square + Area of semicircle

Total Area = 49 + 19.25 = 68.25 cm²

The radius is approximately 3.5 cm and the area is approximately 68.25 cm².