An illustration shows a shape that appears as a square with a semicircle attached on the left side and the top. The side of the square is 2 inches and the diameter of the semicircles is 2 inches. The overlapping lines, which include the top and left side of the square, are marked as dashed lines. Right angle symbols are shown on the top left and bottom right corners of the square.
You are making a card for a friend that looks like a heart shape. Solve for the total area of the shape. Use 3.14 as pi. Round the answer to the nearest hundredth.
(1 point)
Responses
7.14 in.2
7.14 in. squared
16.56 in.2
16.56 in. squared
5.57 in.2
5.57 in. squared
10.28 in.2
10.28 in. squared
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To solve for the total area of the shape, we need to find the individual areas of the square and the two semicircles, and then subtract the overlap area of the semicircles.
The area of the square is calculated by multiplying the length of one side by itself: 2 inches * 2 inches = 4 square inches.
The area of one semicircle is calculated by dividing the area of a full circle (πr^2) by 2: (3.14 * (1 inch)^2) / 2 = 1.57 square inches.
Since there are two semicircles, the total area of the semicircles is 1.57 square inches * 2 = 3.14 square inches.
To find the overlap area, we need to subtract the area of the right triangle formed by the two semicircles from the total area of the semicircles.
The base of the right triangle is the diameter of the semicircle, which is 2 inches. The height of the right triangle is the radius of the semicircle, which is 1 inch. Therefore, the area of the right triangle is (1 inch * 2 inches) / 2 = 1 square inch.
Subtracting the area of the right triangle from the total area of the semicircles gives us 3.14 square inches - 1 square inch = 2.14 square inches.
Finally, to find the total area of the shape, we add the area of the square and the remaining area of the semicircles: 4 square inches + 2.14 square inches = 6.14 square inches.
Rounding to the nearest hundredth, the total area of the shape is approximately 6.14 in².