Question

find the perimeter and area of the following figures (a) and (b). a. the perimeter is _______b. the shaded area is ______

Answer 1

The perimeter of a figure is the length of its outer sides.

In this case, we need to find the perimeter of the semi-circle on the top and the highlighted sides of the two triangles at the bottom.

The perimeter of a semi-circle is given by the following formula:

$P=\pi r$

Where r is the radius of the semi-circle.

The length of the missing sides of the triangles is given by the Pythagorean theorem, like this:

$a=\sqrt[]{b^2^{}+c^2}$

Where b and c are the lengths of the legs of the triangles.

By replacing 8 for r into the first formula, we get:

$P=\pi\times8=8\pi$

By replacing 8 for b and 15 for c into the second equation, we get:

$a=\sqrt[]{8^2+15^2}=\sqrt[]{64+225}=\sqrt[]{289}=17$

Then, we just have to add up the perimeter of the semi-circle and the lengths of the mentioned sides, two times, because there are two triangles, like this:

Perimeter = 8π + 17 + 17 = 8π + 34

Then, the perimeter of figure a equals 8π + 34 meters

The area of this figure can be calculated by summing up the area of the semicircle and the area of the triangle, like this:

The area of the semi-cirlce (red shaded area) is given by the following formula:

$A=\frac{\pi r^2}{2}$

The area of the triangle (yellow shaded area) is given by the following formula:

$A=\frac{b\times h}{2}$

Where b is the length of the base and h is the height of the triangle.

By replacing 8 for r into the first formula, we get:

$A=\frac{\pi\times8^2}{2}=\frac{64\pi}{2}=32\pi$

By replacing 15 for h and 16 for b ( 8 + 8) into the second formula, we get:

$A=\frac{15\times16}{2}=120$

By adding the area of the semi-circle and the area of a triangle, we get:

Area = 32π + 120

Then, the area of the figure equals 32π + 120 square meters