Krishna
0

Step 1: Set up a condition to get the area of the shaded region

Area of shaded shape = area of quadrilateral AMOP - area of sector MOP

Step 2:  Find the area of the quadrilateral

NOTE:  Quadrilateral AMOP is made up of two congruent right triangles (AC perpendicular to PO and AB perpendicular to MO) since P and M are points of tangency. (Note: AP = a/2)

Area of the quadrilateral = 2* area of the right triangle

Skill 1:  Find the area of the right triangle

\frac{1}{2} (b * h)

Skill 2: Find the base and height of the right triangle

Base b = \frac{side}{2}   = \frac{10}{2}

Height h = radius of the inscribed circle = side *\frac{\sqrt{3}}{6}

h = 10 * \frac{\sqrt{3}}{6}

Skill 3: Plugging the base and height values in the formula

\frac{1}{2} *\frac{10}{2}*10 * \frac{\sqrt{3}}{6}

25 \frac{\sqrt{3}}{6}

So, area of the quadrilateral = 2* 25 \frac{\sqrt{3}}{6}

Step 3:Calculate the area of the sector

NOTE: Find the angle of the sector = \frac{360}{3} = 120 \degree

[Step 1: Recall the area of the sector formula

NOTE: \frac{\theta}{360} * \pi r^2

Step 2: Substitute all the values in the formula.

EXAMPLE:   \frac{\theta}{360} * \pi r^2

\frac{120}{360}\ *\ 3.14\ *\ \left(10\ \frac{\sqrt{3}}{6}\right)^2

Step 3: Simplify the equation

EXAMPLE:  \frac{\pi}{3}\cdot\ \left(10\frac{\sqrt{3}}{6}\right)^2

= \frac{50\ \pi}{18}

Step 4:  Calculate the area of the shaded region

NOTE: Area of the shaded region = area of quadrilateral AMOP - area of sector MOP

Area of the shaded region = 2* 25 \frac{\sqrt{3}}{6} - \frac{50 \pi}{18}