Geometry Area and Perimeter HW#72

Geometry

Area and Perimeter ? HW#72

Find the area of the shaded region in each of the following figures. For all

questions, assume that things that look like squares are squares, things that

look like circles are circles, etc.

1:

2:

The radius of each semi-circle must be half the side of the square. Therefore,

the radius is 5. Area of Square = 100cm2 Area of each semi-circle: cm2

Area of Shaded Region = area of square minus both semi-circles: 100 - 25 cm2

3:

Area of Circle: 49 m2 The diagonal of the square is 14

which means each side is 7. Therefore, the area of the square is (7)2 98 m2 The area of the shaded region is the area of the circle minus the area of the square: 49 ? 98 m2 4:

The area of the outer rectangle is: 10 * 8 80 ft2

The area of the inner rectangle is: 5 * 8 40 ft2

The area of the shaded region is: 80 ? 40 40 ft2

Area of outer circle = 64 cm2 Area of inner circle = 25 cm2

Area of shaded region: 64 - 25 39 cm2

5:

6: The shaded square is inscribed

within the larger square.

If we divide this into nine regions, the area of the upper middle section (shaded) is 4m x 5m 20m2. The lower, left, and right shaded sections are also 20m2. The central shaded section is 4m x 4m 16m2. Add all of these shaded sections up and you get 96m2. You can also calculate the area of the larger square (142 196 m2), and then subtract the area of each unshaded corner (52 25 m2), which still yields the same answer of 96m2.

7: This figure consists of 2 concentric circles. If the shaded area is 64 in2 and the smaller circle has a radius of 6 in., what is the radius, in inches of the larger circle?

A square inscribed within a square will result in four congruent triangles. Therefore, each of these unshaded triangles has a longer leg of 4 and a shorter leg of 3. This means that the hypotenuse is 5. Therefore, the area of the shaded square is 52 25. Side Note: This diagram is used to prove the Pythagorean Theorem.

Concentric circles means that the distance between them is consistent all the way around the circles. The area of the shaded region is the area of the larger circle minus the smaller. The smaller circle has a radius of 6 so the area is 36. Use the formula: Area of larger circle - 36 = 64. Solving we get that the area of the larger circle is 100. This means that the radius must be 10 inches.

8:

9:

The diameter of each circle is ? of the square's side. Therefore, the diameters are 8cm and the radii are 4cm, which means that the area of each circle is 16 cm2. Since there are 16 circles, the area of all of them combined is 256 cm2. The area of the square is 1024 cm2. Therefore, the area of the shaded region is: 1024 - 256 cm2

10:

The area of the square is 100 cm2. The diameter of the circle is 10cm so the radius is 5cm. The area of the circle is 25 cm2. Therefore, the area of the shaded region is: 100 - 25 cm2.

Notice that this is the same as question # 1. The only difference is that the circle was broken up into two semi-circles and their positions within the square were exchanged. 11:

The outer semi-circle has a radius of 4cm so the area is half of 16 cm2, which is 8 cm2. The area of each smaller semi-circle is half of 4 cm2, which is 2 cm2. The area of the

shaded region is: 8 ? (2 * 2) 4 cm2

Outer circle: 16 in2 Inner circle: 4 in2 Shaded region: 12 in2

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