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See how a square fits neatly into a circle that’s 8 cm around. It’s all about understanding shapes and space.
The area of the square that can be inscribed in a circle of radius 8 cm is
Area of the square is 128 square cm.
To find the area of the square that can be inscribed in a circle of radius 8 cm, we need to find the side length of the square first.
When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
Let’s denote the side length of the square as “s”. The diagonal of the square is also the diameter of the circle, which is twice the radius, so it’s 2r, where “r” is the radius of the circle.
So, s squared plus s squared equals (2r) squared, by Pythagoras’ theorem.
2s squared equals 4r squared.
Dividing both sides by 2:
s squared equals 2r squared.
Taking the square root of both sides:
s equals square root of 2 times r.
Given that the radius of the circle is 8 cm, we can calculate the side length of the square:
s equals square root of 2 times 8.
s equals 8 times square root of 2 cm.
Now, to find the area of the square, we square the side length:
Area of the square equals (8 times square root of 2) squared.
Area of the square equals 64 times 2.
Area of the square equals 128 square cm.
So, the area of the square that can be inscribed in a circle of radius 8 cm is 128 square cm.
Circle and Its Properties
A circle is a simple yet fundamental shape in geometry, characterized by its continuous, smooth, one-dimensional boundary. Here’s a breakdown of its key properties:
Basic Definition:
- A circle is a collection of all points in a plane equidistant from a fixed point, called the center.
- The distance between any point on the circle and the center is known as the radius.
- The diameter is a straight line segment passing through the center, with its endpoints on the circle. It’s essentially twice the radius (diameter = 2 * radius).
Key Properties:
- Symmetry: A circle exhibits rotational symmetry around its center, meaning it can be rotated at any angle and still look identical.
- Congruence: Circles with equal radii are congruent, meaning they have the same size and shape.
- Circumference: The total length of the circle’s boundary, calculated as circumference = 2 * π * radius.
- Area: The enclosed region within the circle, calculated as area = π * radius^2.
- Chords and Secants: Chords are line segments within the circle connecting two points on its boundary. Secants are lines intersecting the circle at two distinct points.
- Tangents: Lines touching the circle at exactly one point are called tangents. The radius drawn to the point of tangency is always perpendicular to the tangent line.
- Arcs: Portions of the circle’s circumference are called arcs. Their central angles measure the fraction of the full circle they represent.
Points to Remember:
- Circles play a crucial role in various branches of mathematics, including trigonometry, calculus, and analytic geometry.
- They find numerous applications in real-world scenarios, from engineering and physics to design and everyday objects like gears, wheels, and coins.
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