Explanation:
why are you confused ?
the question is to get the area of the shaded (= grey) areas.
all the shapes are clearly combined shapes out of several basic shapes, and in some cases we need to add then, and in some we need to subtract them from each other (to eliminate any white content).
that is really all that needs to be done.
of course, it is important to notice the different metrics used for the lengths, which define the metrics to be used for the areas.
a.
a rectangle and 2 half-circles on the sides (they are together one full circle).
radius is always 1/2 of the diameter of a circle.
and the width of the rectangle is exactly the diameter (= 2×radius) of the circle(s).
grey area :
rectangle : 20×(6×2) = 20×12 = 240 m²
circle : pi×r² = pi×6² = pi×36 = 113.0973355... m²
in total :
240 + 113.0973355... = 353.0973355... m²
b.
a trapezoid minus a circle.
the area of a trapezoid is
(a+b)×h/2
and h (height) is again the diameter of the circle (2×radius).
grey area :
trapezoid : (9 + 15)(4×2)/2 = 24 × 4 = 96 ft²
circle : pi×r² = pi×4² = pi×16 = 50.26548246... ft²
in total :
96 - 50.26548246... = 46.26548246... ft²
c.
3 rectangles.
rectangle1 - rectangle2 + rectangle3.
grey area :
rectangle1 : 14×10 = 140 m²
rectangle2 : 10×6 = 60 m²
rectangle3 : 6×2 = 12 m²
in total :
140 - 60 + 12 = 92 m²
d.
a circle minus a square.
the only specialty here is the diameter and radius of the circle : the diagonal of the inner square.
we get the diagonal of the inner square via Pythagoras, because the diagonal makes 2 right-angled triangles out of the square.
diagonal² = 4² + 4² = 16+16 = 32
diagonal = sqrt(32)
radius = diagonal/2 = sqrt(32)/2 = sqrt(32/4) = sqrt(8)
grey area :
square : 4×4 = 16 cm²
circle : pi×r² = pi×(sqrt(8))² = pi×8 = 25.13274123... cm²
in total :
25.13274123... - 16 = 9.132741229... cm²
is there still anything unclear ?
please let me know.
but as you can see there is no need to panic. just use basic math and common sense.