ACDE is a Paraltelogram. DEF is a sector. ABL is a right trianghe. Solve for the area of the composite figure.

Question

ACDE is a Paraltelogram. DEF is a sector. ABL is a right triangle. Solve for the area of the composite figure.

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Answer 1

To find the area of the composite figure, sum the area of the triangle ABC, the area of the parallelogram ACDF, and the area of the sector FDE.

1) Find the area of the triangle ABC

Area of a triangle is given by the formula: A = 1/2 * b * h

b = 4cm

h = 3cm

A = 1/2 * 4cm * 3cm = 6 sq cm

2) Find the area of the parallelogram ACDF

Area of a parallelogram is given by the following formula:

A = b*h, where b is the base, and h is the line drawn perpendicular to two parallel edges

b = 20 cm

h = 4 cm (given to use by the triangle)

A = 20cm * 4cm = 80 sq cm

3) Find the area of the sector FDE

The area of the sector is the area of the fraction of the circle it makes, based off the labeled angle.

A_sector = fraction of the circle * A_circle

The area of the whole circle can be given by:

A_circle = πr²

What is the radius?

In this case, the radius is the same the edge of the parallelogram that wasn't labeled. By definition of a parallelogram, edge DF is the same length as AC. AC is also the hypotenuse of the triangle. Because ABC is a right triangle with lengths 3 cm and 4 cm, we can use the 3-4-5 Pythagorean triple to ascertain that the hypotenuse is 5cm. We could also use the Pythagorean theorem for this 3² + 4² = c²

So the whole circle's area A_circle = π(5cm)² = 25π sq cm

Thus the sector's area is A_{sector =} 35/360*A_circle =35/360*(25π sq cm) = 875π/360 sq cm

4) Add these all together

A_composite = A_triangle + A_{parallelogram} + A_sector

A_composite = 6 sq cm + 80 sq cm + 875π/360 sq cm

A_composite ≅ 93.59 sq cm

So the area of the entire composite figure is about 93.59 sq cm

Answer 2

Calculate, label all segments, use area formula to answer