We must calculate the area of the large semicircle and subtract that area by the area of the two semicircles since they are taking up a fraction of the large semicircle’s area.
Let’s first calculate the area of the two semicircles.
Statement:
A=πr²
Reason: Area of a circle formula
Statement:
A=1/2•πr²
Reason: a semicircle is a half circle, thus it has half the area of a regular circle, hence why 1/2 is multiplying the area formula.
Statement:
A1+A2=(1/2•πr²)+(1/2•πr²)
Reason: A1=area of semicircle 1 and A2=area of semicircle 2. We must know the total area of the two semicircles because they lie within the large semicircle and are taking up some of its area.
Statement:
A1+A2=2(1/2•πr²)
Reason: by definition of multiplication, let “a” be any real number, and we see that a+a=2a. Because the two semicircles contain equal diameters, their radii must be the same, and thus the two semicircles are congruent. So, adding the equal area of the two semicircles is the same as multiplying the area of one semicircle by 2. It’s important to note that the factor of 2 will cancel out the 1/2, forming a whole circle. This should make sense because the area of one semicircle plus the area of a second, congruent semicircle forms an entire circle.
Statement:
Let s=area of semicircles 1 + 2:
s=2(1/2•πr²)
Input the values into the formula. Note that the diameter is double the radius, so the radius is 1/2 the diameter. We are given to approximate pi to 3.142, so we’ll replace pi with 3.142:
s=2(1/2•(3.142)(15/2)²)
Simplify:
s=2(1/2•3.142(7.5)²)
s=2(1/2•3.142•56.25)
s=2(88.36875)
s=176.7375cm^2
Now that we have the area of the two semicircles, we can calculate the area of the large semicircle and subtract its area by the two semicircles. We know the radius of the large semicircle is 15cm because it’s the diameter of the smaller semicircles.
Statement:
A=1/2(3.142•r²)
Reason: area of semicircle formula
Statement:
A=1/2(3.142•(15)²)
Reason: substituting values into formula
Statement:
A=1/2(3.142•225)
A=1/2(706.95)
A=706.95/2
A=353.475cm^2
Reason: simplification
Statement:
A—s=353.475–176.7375
Where A=large semicircle area and s=two smaller semicircle’s area
Total area: 176.7375cm^2