The figures in (a) and (b) below are made up of semicircles and quarter circles; the figure in (c) shows a quarter of a circle in a square. Find the area and the perimeter of each figure in terms of π.
sorry - not much specific information here.
Allie
How dumb can u be
To find the area and perimeter of each figure, let's analyze them one by one:
(a) Figure made up of semicircles and quarter circles:
To find the area, we need to calculate the sum of the areas of the semicircles and quarter circles.
The formula for the area of a semicircle is:
A = πr^2/2
The formula for the area of a quarter circle is:
A = πr^2/4
Let's say the radius of the semicircles and quarter circles is 'r'.
In figure (a), there are 4 semicircles and 4 quarter circles.
The total area of the figure will be:
A = 4 × (πr^2 / 2) + 4 × (πr^2 / 4)
= 2πr^2 + πr^2
= 3πr^2
To find the perimeter, we need to calculate the sum of the circumferences of the semicircles and quarter circles.
The formula for the circumference of a semicircle is:
C = πr + 2r
The formula for the circumference of a quarter circle is:
C = (πr) / 2 + 2r
In figure (a), there are 4 semicircles and 4 quarter circles.
The total perimeter of the figure will be:
P = 4 × (πr + 2r) + 4 × ((πr) / 2 + 2r)
= 4πr + 8r + 2πr + 8r
= (6π + 16) r
(b) Figure made up of semicircles and quarter circles:
The analysis for figure (b) will be similar to figure (a), as it also consists of semicircles and quarter circles.
The total area of the figure will be:
A = 3πr^2
The total perimeter of the figure will be:
P = (6π + 12) r
(c) Figure with a quarter circle in a square:
To find the area of this figure, we need to calculate the area of the quarter circle and the area of the square.
The formula for the area of a square is:
A = s^2
Let's say the side length of the square is 's'.
The area of the quarter circle will be:
A_quarter = πr^2/4
Since the quarter circle is inscribed in the square, the radius (r) will be half of the side length of the square.
So, r = s/2
The total area of the figure will be:
A = π(s/2)^2/4 + s^2
= πs^2/16 + s^2
= (π/16 + 1) s^2
To find the perimeter, we need to calculate the sum of the perimeter of the square and the circumference of the quarter circle.
The formula for the perimeter of a square is:
P = 4s
The formula for the circumference of a quarter circle is:
C = (πr)/2 + 2r
The perimeter of the figure will be:
P = 4s + ((πs/2)/2 + 2s)
= 4s + (πs/4 + 2s)
= (9π/4 + 6) s
Therefore, the area and perimeter of each figure in terms of π are as follows:
(a) Area = 3πr^2
Perimeter = (6π + 16) r
(b) Area = 3πr^2
Perimeter = (6π + 12) r
(c) Area = (π/16 + 1) s^2
Perimeter = (9π/4 + 6) s
To find the area and perimeter of each figure, we will break them down into their individual components and then calculate accordingly. Let's start with figure (a).
Figure (a):
This figure is made up of semicircles and quarter circles. Let's calculate the area and perimeter of each component and then add them up.
1. Semicircle:
The formula for the area of a semicircle is given by A = (π * r^2) / 2, where r is the radius.
The formula for the perimeter of a semicircle is given by P = π * r + 2r, where r is the radius.
2. Quarter circle:
The formula for the area of a quarter circle is given by A = (π * r^2) / 4, where r is the radius.
The formula for the perimeter of a quarter circle is given by P = (π * r) + (2r), where r is the radius.
Now, let's calculate the area and perimeter of each component:
Area of semicircle = (π * r^2) / 2
= (π * (radius of semicircle)^2) / 2
Perimeter of semicircle = π * r + 2r
= π * (radius of semicircle) + 2 * (radius of semicircle)
Area of quarter circle = (π * r^2) / 4
= (π * (radius of quarter circle)^2) / 4
Perimeter of quarter circle = (π * r) + (2r)
= π * (radius of quarter circle) + 2 * (radius of quarter circle)
Once you have calculated the areas and perimeters of each individual component, you can add them up to find the final area and perimeter of figure (a).
Repeat the same process for figures (b) and (c), which are made up of semicircles, quarter circles, and a square for figure (c).
Remember to substitute the radius in the formulas with the actual values given in the figure.
By following this process, you will be able to find the area and perimeter of each figure in terms of π.