Calculating the Area of the Circle

Oct 13, 2025 by thesweetdesigns.com 48 views

Introduction

When a circle is inscribed within a rectangle, the circle touches the sides of the rectangle at the midpoint of each side. This creates a shaded region, which is the area between the circle and the rectangle. In this article, we will explore how to find the area of the shaded region when a circle with a radius of 3 cm is inscribed inside a rectangle with dimensions 8 cm x 7 cm.

Understanding the Problem

To find the area of the shaded region, we need to understand the relationship between the circle and the rectangle. The circle is inscribed within the rectangle, meaning that it touches the sides of the rectangle at the midpoint of each side. This creates four right-angled triangles, each with a base and height equal to the radius of the circle.

Calculating the Area of the Circle

The area of a circle is given by the formula:

A = πr^2

where A is the area of the circle, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

In this case, the radius of the circle is 3 cm, so we can plug this value into the formula to find the area of the circle:

A = π(3)^2 A = 3.14 x 9 A = 28.26

Calculating the Area of the Rectangle

The area of a rectangle is given by the formula:

A = length x width

In this case, the length of the rectangle is 8 cm and the width is 7 cm, so we can plug these values into the formula to find the area of the rectangle:

A = 8 x 7 A = 56

Calculating the Area of the Shaded Region

To find the area of the shaded region, we need to subtract the area of the circle from the area of the rectangle. This will give us the area of the four right-angled triangles that are formed by the circle and the rectangle.

A_shaded = A_rectangle - A_circle A_shaded = 56 - 28.26 A_shaded = 27.74

Rounding the Final Answer

The final answer should be rounded to the nearest hundredth. Therefore, we can round the area of the shaded region to 27.74.

Conclusion

In this article, we have explored how to find the area of the shaded region when a circle with a radius of 3 cm is inscribed inside a rectangle with dimensions 8 cm x 7 cm. We have used the formulas for the area of a circle and a rectangle to find the area of the shaded region, and have rounded the final answer to the nearest hundredth.

Calculating the Area of the Shaded Region Using a Different Method

Another way to find the area of the shaded region is to use the formula for the area of a rectangle and subtract the area of the four right-angled triangles that are formed by the circle and the rectangle.

The area of each right-angled triangle is given by the formula:

A_triangle = (base x height) / 2

In this case, the base and height of each triangle are equal to the radius of the circle, which is 3 cm. Therefore, we can plug these values into the formula to find the area of each triangle:

A_triangle = (3 x 3) / 2 A_triangle = 9 / 2 A_triangle = 4.5

Since there are four right-angled triangles, we can multiply the area of each triangle by 4 to find the total area of the triangles:

A_triangles = 4 x 4.5 A_triangles = 18

Now, we can subtract the area of the triangles from the area of the rectangle to find the area of the shaded region:

A_shaded = A_rectangle - A_triangles A_shaded = 56 - 18 A_shaded = 38

However, this is not the correct answer. The correct answer is 27.74. This is because the area of the shaded region is actually the area of the rectangle minus the area of the circle, not the area of the rectangle minus the area of the four right-angled triangles.

Final Answer

The final answer is: $\boxed{27.74}$