Answers:
10. 28.3 cm²
11. 7.3 cm²
12. P: 32 yds, A: 76.8 sq yds
13. P: 64 units, A: 220.8 units
14. 50.3 cm²
15. 132.7 cm²
16. 9.7 in
17. 5.9 cm
18. 324.08 sq inches.
19. 39 cm²
20. 36.6 cm²
Explanation:
10. Find the area of the shaded region.
It's a circle, let's first calculate its area, with the formula A = π * r²
A = π * 6² = 36 π = 113.1 cm²
The shaded area has a 90 degrees angle... so it's 1/4 of the whole circle.
The area of the shaded area of this circle is then: 113.1 * 1/4 = 28.275 cm²
Rounded to the tenths, that's of course 28.3 cm²
11. Find the area of the shaded region.
Again, let's first find the area of the complete circle, with A = π * r²
A = π * 4² = 16π = 50.27 cm²
Now, let's find the angle for the shaded area.
A circle has 360 degrees, and we know the angles of 2 segments... so we can easily find the missing angle:
360 - 203 - 105 = 52
The shaded area has an arc of 52 degrees, so we multiply the area of the full circle by 52/360 to get the area of that shaded area:
(50.27 * 52) / 360 = 7.26 cm², rounded to the tenths: 7.3 cm²
12. Find perimeter AND area of this regular polygon.
The figure is a regular octagon (8 sides).
To calculate its perimeter, it's simply 8 times one side, so: 8 * 4 yd = 32 yds
For the area, you can view an octagon as 8 triangles joined together. In this case, we have a base of 4 yds and a height of 4.8 yds, so the area of each triangle is: (4 * 4.8) /2 = 9.6, the total area of the octagon is then 8 * 9.6 = 76.8 yds
13. Find perimeter AND area of this regular polygon.
The figure is a regular octagon (8 sides).
To calculate its perimeter, it's simply 8 times one side, so: 8 * 8 = 64 units
For the area, you can view an octagon as 8 triangles joined together. In this case, we have a base of 4 yds and a height of 4.8 yds, so the area of each triangle is: (8 * 6.9) /2 = 27.6, the total area of the octagon is then 8 * 27.6 = 220.8 units
14. Find the area of a circle of radius = 4 cm
We have a circle with a radius of 4 cm, we need to find its area.
The area of a circle is obtained by the formula: A = π * r²
We already have the value of r, so we will input it in the formula:
So, we'll have A = π * 4² = 16 * π = 50.27 cm²
Rounded to the tenths: 50.3 cm²
15. Find the area of a circle of diameter 13 cm
We have a circle with a diameter of 13 cm, we need to find its area. To use the formula for the area, we need the radius, not the diameter. Since the diameter is 13 cm, the radius is 6.5
The area of a circle is obtained by the formula: A = π * r²
So, we'll have A = π * 6.5² = 42.25 * π = 132.73 cm²
Rounded to the tenths: 132.7 cm²
16. Find the diameter of a circle with an area of 75 in².
We just used the area formula for a circle, based on the radius. We'll process it in reverse from the area to get the radius... which will give us the diameter, so r² = A / π
We then have: r² = 75 / π = 23.87
So, r = 4.88 (square root of 23.87), which we double to get the diameter: 9.76 in, rounded to 9.7 in.
17. Find the radius of a circle with an area of 108 cm².
We just used the area formula for a circle, based on the radius. We'll process it in reverse from the area to get the radius, so r² = A / π
If we input the area given in the question into the formula, we have: r² = 108 / π = 34.38 cm
So, r = 5.86 (square root of 34.38), rounded to 5.9 cm.
18. Find the area
We have here a half-circle with a triangle.
We just used the formula for the area of a circle, and we know the radius of that half-circle: 8 inches.
A = π * 8² = 64 * π = 201.06 sq inches, for the whole circle.
But we only have half of it, so 201.06 / 2 = 100.08 sq inches.
Now the triangle. To calculate the area of a triangle, you multiply its base by its height, then you divide by 2, so:
A = 16 x 28 = 224 sq inches.
Which we add to the half-circle area (100.08 sq inches) to get 324.08 sq inches. So 324.1 sq inches once rounded to the tenths.
19. Find the area
We have a triangle on top of a rectangle.
Area of a triangle: (base * height) / 2.
Area of a rectangle: base * height
So, for the triangle: A = (6 * 3) / 2 = 9 cm²
And for the rectangle: A = 6 * 5 = 30 cm²
Total: 9 + 30 = 39 cm²
20. Find the area
We have a rectangle and a half circle. The radius of the half-circle is: 2 cm (half of the height of the rectangle).
Area of the rectangle: base * height, so 6 * 4 = 24 cm²
Area of the circle: π * r² = π * 2² = 4π = 12.56 cm²
Total for the figure: 24 + 12.56 = 36.56, or 36.6 cm² once rounded.