Find the area of a square inscribed in a circle of radius 10cm
200
The diagonal of a square inscribed in the given circle measures the same as the diameter of the circle, 2*10=20 cm.
Since the diagonal cuts the square into two right triangles, we consider one of the two right triangles.
Let the side of the square = x,
then area of square = x²
From Pythagoras theorem,
x² + x² = 20²
we conclude that
area of square = x² = 20²/2
To find the area of a square inscribed in a circle, we need to find the length of the sides first.
In this case, the diameter of the circle is twice the radius, so it is 2 x 10 cm = 20 cm.
Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.
Therefore, the length of each side of the square is the diagonal divided by the square root of 2.
The formula for the area of a square is side^2.
Let's calculate it step-by-step:
Step 1: Calculate the length of each side of the square.
Side length = Diameter / √2
Side length = 20 cm / √2
Side length ≈ 14.142 cm (rounded to three decimal places)
Step 2: Calculate the area of the square.
Area of square = side length^2
Area of square ≈ (14.142 cm)^2
Area of square ≈ 200 cm^2 (rounded to three decimal places)
Therefore, the area of the square inscribed in a circle with a radius of 10 cm is approximately 200 cm^2.
To find the area of a square inscribed in a circle, we can use the fact that the diagonal of the square is equal to the diameter of the circle. Here's how we can find the area:
1. Start by finding the length of the diagonal of the square. Since the diagonal is equal to the diameter of the circle, which is 2 times the radius, the length of the diagonal is 2 * 10 cm = 20 cm.
2. To find the length of the sides of the square, we can use the Pythagorean theorem. The diagonal of the square divides it into two right-angled triangles. Let's call one of the legs of the triangle x. In this case, x represents half the length of the side of the square.
Using the Pythagorean theorem, we have: x^2 + x^2 = 20^2. Simplifying this equation, we get: 2x^2 = 400. Dividing both sides by 2, we have: x^2 = 200. Taking the square root of both sides, we get: x = sqrt(200) cm.
3. Now that we know the length of one side of the square, we can easily find its area. The area of a square is given by the formula A = side^2. Substituting x with sqrt(200) cm, we have A = (sqrt(200))^2 = 200 cm^2.
Therefore, the area of the square inscribed in a circle with a radius of 10 cm is 200 square centimeters.