Answer:
top: 2.6 units²
bottom: 2.83 units²
Step-by-step explanation:
so you need to know a couple of things
1) the inscribed polygons are made up of isoceles triangles (the top one has 6 and the bottom one has 8 triangles)
2) we know the length of each side of the isosceles triangles (given as r = 1 for all triangles)
3) we know that the entire circle is 360 degrees and therefore the angle of segment of the pie is 360° ÷ number of triangles in each circle
for the top, each segment has an angle of 360°÷6 = 60°
for the bottom, each segment has an angle of 360° ÷ 8 = 45°
4) We can find the area of a triangle if we know two sides and the angle between the two known sides using the following formula
Area = (1/2) x (Length 1) x (Length 2) x sin (angle between them)
In our case, we know that the length of each side is the same length 1 and the angle between them is either 60° (for top) and 45° (for bottom)
hence,
Area of each triangle = (1/2) x 1 x 1 x sin (angle)
= (1/2) sin (angle)
Area of each polygon= area of each triangle x number of triangles
= (1/2) sin (angle) x number of triangles.
For the top, each segment angle is 60° and there are 6 triangles.
Area of top polygon = (1/2) sin (60°) x 6 = 2.6 units²
For the bottom, each segment angle is 45° and there are 8 triangles.
Area of top polygon = (1/2) sin (45°) x 8 = 2.83 units²
