Imagine you have a square piece of paper. Each side of the square is 10 centimeters long. Now, starting from one corner of the square, you draw a quarter circle that perfectly fits into that corner. This curve reaches the two edges of the square next to the corner, forming a smooth curved shape inside the square.
Now, you are asked:
👉 What is the area of the part of the square that is not covered by the quarter circle?
🟠 How to Solve (step by step):
Find the area of the whole square:
Since each side is 10 cm, the area of the square is
10
×
10
=
100
square centimeters
.
10×10=100square centimeters.
Find the area of the quarter circle:
A quarter circle means one-fourth of a full circle.
The formula for the area of a full circle is
𝜋
𝑟
2
,
where
𝑟
=
radius
.
πr
2
, where r=radius.
Here, the radius is also 10 cm (because it goes from the corner to the edges of the square).
So the area of the quarter circle is
1
4
×
𝜋
×
10
2
=
1
4
×
𝜋
×
100
=
25
𝜋
square centimeters
.
4
1
×π×10
2
=
4
1
×π×100=25πsquare centimeters.
(Which is approximately 78.54 cm² when using π ≈ 3.1416)
Subtract to find the shaded area:
The shaded area is the part of the square not covered by the quarter circle.
So we subtract the area of the quarter circle from the area of the square:
100
−
25
𝜋
≈
100
−
78.54
=
21.46
square centimeters
.
100−25π≈100−78.54=21.46square centimeters.
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答案文本
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We start with a square piece of paper where each side is 10 centimeters long. From one corner of the square, we draw a quarter circle that fits perfectly into that corner. The quarter circle touches the two adjacent edges of the square. Our goal is to find the area of the part of the square that is NOT covered by this quarter circle.
First, we need to find the area of the whole square. The formula for the area of a square is side times side. Since each side of our square is 10 centimeters, we calculate 10 times 10, which equals 100 square centimeters. This grid visualization helps us see that the square contains 100 unit squares, each representing 1 square centimeter.
Now let's understand what a quarter circle is. A quarter circle is exactly one-fourth of a full circle. The area of a full circle is pi times radius squared. Therefore, the area of a quarter circle is one-fourth of that, which equals one-fourth times pi times radius squared. In our problem, the radius of the quarter circle is 10 centimeters, the same as the side length of the square.
Now let's calculate the area of the quarter circle. We use the formula: one-fourth times pi times radius squared. Substituting our radius of 10 centimeters, we get one-fourth times pi times 10 squared, which equals one-fourth times pi times 100. This simplifies to 25 pi square centimeters. Using pi approximately equal to 3.1416, this gives us about 78.54 square centimeters.