What is the perimeter of a square inscribed in a circle of radius 5 cm?
20√2 cm
Let's find the perimeter of a square inscribed in a circle. We are given that the radius of the circle is 5 cm.
Understanding Inscribed Shapes
When a square is inscribed in a circle, it means that all four vertices (corners) of the square lie on the circumference of the circle. In this arrangement, the diagonal of the square is also the diameter of the circle.
- Radius of the circle (r): 5 cm
- Diameter of the circle (d): The diameter is twice the radius. So, \(d = 2 \times r = 2 \times 5\) cm.
- \(d = 10\) cm
Since the diagonal of the inscribed square is equal to the diameter of the circle, the diagonal of the square is 10 cm.
Finding the Side Length of the Square
Let the side length of the square be 's'. In a square, the relationship between the side length (s) and the diagonal (d) is given by the Pythagorean theorem. If we consider one of the right-angled triangles formed by the diagonal and two sides of the square, we have:
\(s^2 + s^2 = d^2\)
\(2s^2 = d^2\)
We know the diagonal \(d = 10\) cm. Let's substitute this value:
\(2s^2 = 10^2\)
\(2s^2 = 100\)
Now, we solve for \(s^2\):
\(s^2 = \frac{100}{2}\)
\(s^2 = 50\)
To find the side length 's', we take the square root of 50:
\(s = \sqrt{50}\)
We can simplify \(\sqrt{50}\) by factoring 50 as \(25 \times 2\):
\(s = \sqrt{25 \times 2}\)
\(s = \sqrt{25} \times \sqrt{2}\)
\(s = 5\sqrt{2}\) cm
So, the side length of the square is \(5\sqrt{2}\) cm.
Calculating the Perimeter of the Square
The perimeter of a square is the sum of the lengths of its four sides. Since all sides of a square are equal, the perimeter is 4 times the side length.
Perimeter \(P = 4 \times s\)
Substitute the side length \(s = 5\sqrt{2}\) cm:
\(P = 4 \times (5\sqrt{2})\)
\(P = (4 \times 5) \times \sqrt{2}\)
\(P = 20\sqrt{2}\) cm
The perimeter of the square inscribed in the circle with a radius of 5 cm is \(20\sqrt{2}\) cm.
Summary of Steps
- Identify the relationship between the circle's diameter and the square's diagonal when the square is inscribed in the circle.
- Calculate the circle's diameter (which equals the square's diagonal) using the given radius.
- Use the diagonal to find the side length of the square using the Pythagorean theorem or the formula \(s = d/\sqrt{2}\).
- Calculate the perimeter of the square using the formula \(P = 4s\).
Here is a summary of the values:
| Measurement | Value |
|---|---|
| Circle Radius | 5 cm |
| Circle Diameter (Square Diagonal) | 10 cm |
| Square Side Length | \(5\sqrt{2}\) cm |
| Square Perimeter | \(20\sqrt{2}\) cm |
Revision Table: Inscribed Square Geometry
| Concept | Formula / Relationship | Notes |
|---|---|---|
| Circle Diameter (d) | \(d = 2r\) | 'r' is the circle radius |
| Square Diagonal (ds) in Inscribed Square | \(d_s = d_{circle}\) | Diagonal equals circle diameter |
| Square Side (s) from Diagonal (ds) | \(s = \frac{d_s}{\sqrt{2}}\) or \(d_s = s\sqrt{2}\) | From Pythagorean theorem: \(s^2+s^2 = d_s^2\) |
| Square Perimeter (P) | \(P = 4s\) | Sum of the four equal sides |
Additional Information: Related Geometric Concepts
Understanding inscribed shapes helps in solving many geometry problems. Here are a few related concepts:
- Circumscribed Square: A square is circumscribed around a circle if all four sides of the square are tangent to the circle. In this case, the side length of the square is equal to the diameter of the circle.
- Area of Inscribed Square: If the side length is \(s\), the area is \(s^2\). Using the diagonal \(d = 2r\), \(s = d/\sqrt{2} = 2r/\sqrt{2} = r\sqrt{2}\). Area \(= (r\sqrt{2})^2 = 2r^2\).
- Area of Circumscribed Square: If the side length is \(s = d = 2r\), the area is \(s^2 = (2r)^2 = 4r^2\). Notice the area of a circumscribed square is twice the area of an inscribed square for the same circle.
- Inscribed Circle in a Square: If a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square.
Important Questions from Plane Figures
If length of a rectangle is increased to its three times and breadth is decreased to its half, then the ratio of the area of given rectangle to the area of new rectangle is:
The width of the path around a square field is 4.5 m and its area is 105.75 m 2. Find the cost of fencing the field at the rate of Rs. 100 per meter.
What is the area of the square (in cm 2) whose vertices lie on a circle of radius 5 cm?
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
The perimeter of a circular lawn is 1232 m. There is 7 m wide path around the lawn. The area (in m 2) of the path is:
Take \(\left(\pi=\frac{22}{7}\right)\)
