Diyu P. answered • 03/31/25
Patient MD, Ivy-League Geometry Tutor with 10+ years of experience
To find the area of the composite figure, you can add the areas of each shape.
1) Find the area of the sector AGB.
Think of a sector as a fraction of a circle. In this case, the radius of this circle is AG = 20. The fraction of the circle would be 47/360, since 360 is the total number of degrees in the circle. So the area of this can be represented by the fraction * area of the whole circle.
area of a circle = π*r2
area of the sector = fraction * π*r2
A = 47/360 * pi*202 = 47*400/360π = 47*10*π/9 = 470π/9
2) Find the area of the triangle GBF
a) Find the base
The triangle here is a right triangle, where the base is GF, so b = 12.
b) Find the height
Since the hypotenuse of this right triangle is the same as the radius of the sector, we know the hypotenuse is 20. We can use this to calculate the height since this is a right triangle. Notice that 12 and 20 are multiples of the 3-4-5 Pythagorean triple. We had to multiply 3 and 5 by 4 to get our 12 and 20. Therefore, the height of the triangle is 4*4 = 16. An alternative to this would be to use the Pythagorean theorem to find the height BF, where 122 + BF2 = 202.
c) Put it together to find the area
The area of the triangle can be found using the formula A=1/2×b×h
A = 1/2×12×16=96
3) Find the area of the square
The side of this square is shared by the height of the left triangle. The side is 16.
The area of a square can be found using the formula A = s2
A = 162=256
4) Notice that the triangle ECD shares the same side with the square and also has a base 12. Therefore, its b and h are the same as triangle GBF. Their area is the same
5) Add all the areas together
Area of the composite figure = area of the sector + area of triangle GBF + area of square BCEF + area of triangle ECD
Area of the composite figure = 470π/9 + 96 + 256 + 96 = 470π/9 + 448 ≅ 612.05
So the total area of the composite figure is about 612.05 sq units
