Question

Find the area of the shaded region of circle R below. Round your answer to the nearest tenth if necessary.

Answer 1

71.4 units²

Explanation:

To find:-

• The area of the shaded region.

Answer:-

We can divide the shaded circle into parts , viz a triangle and a sector . We can use the formula to find out the area of the sector which is ,

`\longrightarrow \boxed{\rm{Area} =(\theta)/(2\pi)* \pi r^2 }\\`

where,

• 
  `\theta` is the angle substended by the arc at the centre in radians.
• 
  `r` is the radius of the circle.

The angle substended by the traingle at the centre is 90° . In radians it would be π/2 as π rad = 180⁰ .

Now we can see that the angle substended by the arc would be ,

`\longrightarrow \theta = 2\pi -(\pi)/(2)\\`

`\longrightarrow \theta = (4\pi-\pi)/(2) \\`

`\longrightarrow \theta =\boxed{(3\pi)/(2)} \\`

Now substitute the respective values, in the given formula as ,

`\longrightarrow A =((3\pi)/(2))/(2\pi)*\pi (5^2) \\`

`\longrightarrow A = (3)/(4)* (22)/(7)* 25 \\`

`\longrightarrow \red{ A = 58.93 \ \rm{units}^2}\\`

`\rule{200}2`

Finding the area of triangle:-

The given triangle is a right angled triangle with two sides as 5 . So we know that;

`\longrightarrow \boxed{ \rm{Area}=(1)/(2) bh } \\`

where ,

• 
  `b` is the base = 5 units
• 
  `h` is the height = 5 units

On substituting the respective values, we have;

`\longrightarrow A = (1)/(2)* (5\cdot 5) \\`

`\longrightarrow A = (25)/(2) \\`

`\longrightarrow \red{ A = 12.5 \rm{units}^2} \\`

Hence the total area would be ,

`\longrightarrow A = (58.93 + 12.5) u^2 \\`

`\longrightarrow A = 78.43 \ u^2 \\`

Rounding off to nearest tenth would give us ,

`\longrightarrow \underline{\underline{ \red{ \rm Area_(shaded \ region) = 71.4 \ units^2}}} \\`

This is the required area of the shaded region.

Answer 2

Step 1: Find the area of the triangle

Step 2: Find the area of the shaded area outside the triangle

Step 3: Add them up

Step 1:

Because we know what the radius is, we can also assume that both sides that are needed to find the area are the same because the radiuses are also the same. This means:

A= 5x5 = 25/2 = 12.5cm*2


Step 2:

We can also see that the shaded area outside of the triangle is 3/4 of a circle, therefore we need to find the area of that. This means:

A = pie x radius*2
A= 3.14 x 5*2
A= 78.5/4 (to find the area of 1 quarter)
A= 19.63 x 3 (to find the 3 quarters)
A= 58.59 cm

Step 3:

Now we add them up to find the total area:

58.59+12.5= 71.39