Finding the Area of Shaded Regions

Area of Outer minus Inner Shape

Finding the area of a specific shaded zone by subtracting the area of the smaller inner shape from the area of the larger outer shape.

SA = OA - IA

SA: Shaded Area, OA: Outer Area (total boundary region), IA: Inner Area (unshaded geometry inside to be removed).

Example

A shaded region is formed by a small square with a side of 3 inches inside a large square with a side of 10 inches. Find the shaded area.

Show solution

Step 1: Calculate the area of the outer square (OA): $$10 \times 10 = 100$$ square inches

Step 2: Calculate the area of the inner unshaded square (IA): $$3 \times 3 = 9$$ square inches

Step 3: Subtract the inner area from the outer area (SA = OA - IA): $$100 - 9 = 91$$ square inches

Practice problems

Problem 1
A rectangular lawn measures 12 meters by 8 meters. A rectangular swimming pool measuring 6 meters by 4 meters is built inside it. What is the remaining area of the lawn?

Show solution

Answer: 72 square meters

Step 1: Compute the outer rectangle area (OA): $$12 \times 8 = 96$$
Step 2: Compute the inner rectangle area (IA): $$6 \times 4 = 24$$
Step 3: Subtract the inner area from the outer area (SA = OA - IA): $$96 - 24 = 72$$ square meters
Problem 2
A large rectangular piece of metal sheet is 15 inches long and 10 inches wide. A smaller 5 inch by 4 inch rectangle is cut out from the center. Find the area of the remaining metal sheet.

Show solution

Answer: 130 square inches

Step 1: Calculate the total outer sheet area (OA): $$15 \times 10 = 150$$
Step 2: Calculate the inner cut-out area (IA): $$5 \times 4 = 20$$
Step 3: Subtract the cut-out area from the total area (SA = OA - IA): $$150 - 20 = 130$$ square inches
Problem 3
A square wooden frame has an outer side length of 8 cm. The empty square opening inside has a side length of 5 cm. What is the area of the wooden frame?

Show solution

Answer: 39 square cm

Step 1: Find the outer total square area (OA): $$8 \times 8 = 64$$
Step 2: Find the inner empty square area (IA): $$5 \times 5 = 25$$
Step 3: Subtract the inner area from the outer area (SA = OA - IA): $$64 - 25 = 39$$ square cm

Remember

Calculate the complete area of each shape independently using its specific formula before performing the subtraction.

Shaded Regions with Polygons and Circles

Finding the shaded area when the problem combines a polygon (like a square or rectangle) with a circle or circle parts.

SA = PA - CA

SA: Shaded Area, PA: Polygon Area (boundary shape), CA: Circle Area (inner circular cutout).

Example

A square wooden board has a side length of 6 inches. A circle with a diameter of 6 inches is painted inside, touching all four edges. Find the area of the unpainted corners.

Show solution

Step 1: Find the outer polygon area (PA): $$6 \times 6 = 36$$ square inches

Step 2: Find the inner circle radius $$(\frac{6}{2} = 3)$$ and circle area (CA): $$3.14 \times 3 \times 3 = 28.26$$ square inches

Step 3: Subtract the circle from the square (SA = PA - CA): $$36 - 28.26 = 7.74$$ square inches

Practice problems

Problem 4
A circular metal plate with a radius of 10 cm has a square hole drilled out of its center. If the square hole has a side length of 4 cm, find the area of the remaining metal plate.

Show solution

Answer: 298 square cm

Step 1: Calculate the total outer circle area (CA): $$3.14 \times 10 \times 10 = 314$$
Step 2: Calculate the inner square cutout area (PA): $$4 \times 4 = 16$$
Step 3: Subtract the inner shape from the outer shape (SA = CA - PA): $$314 - 16 = 298$$ square cm
Problem 5
A rectangle has a length of 20 cm and a width of 10 cm. Inside it, two identical circles are drawn, each with a diameter of 10 cm, filling the rectangle completely from left to right. What is the area of the shaded space outside the circles?

Show solution

Answer: 43 square cm

Step 1: Calculate the outer rectangle area (PA): $$20 \times 10 = 200$$
Step 2: Calculate the radius of one circle $$(\frac{10}{2} = 5)$$ and the total area of both circles (CA): $$2 \times (3.14 \times 5 \times 5) = 2 \times 78.5 = 157$$
Step 3: Subtract the circles' total area from the rectangle area (SA = PA - CA): $$200 - 157 = 43$$ square cm
Problem 6
A target sheet consists of a square with side lengths of 12 inches. A red circular bullseye with a radius of 3 inches is located in the center. Find the area of the target sheet outside the bullseye.

Show solution

Answer: 115.74 square inches

Step 1: Calculate the outer square area (PA): $$12 \times 12 = 144$$
Step 2: Calculate the inner circle bullseye area (CA): $$3.14 \times 3 \times 3 = 28.26$$
Step 3: Subtract the inner area from the total area (SA = PA - CA): $$144 - 28.26 = 115.74$$ square inches

Fractional Shaded Areas (Sectors and Segments)

Finding the area of a shaded region that represents a specific fraction or slice of a circle, such as a sector or a segment.

SecA = $$(\frac{n}{360}) \times \pi \times r^2$$

SecA: Sector Area, n: central angle in degrees, r: radius of the circle, pi: approximately 3.14.

Example

A circular pie has a radius of 6 cm. A slice with a central angle of 60 degrees is shaded. Find the area of this slice.

Show solution

Step 1: Convert the sector angle into a fraction $$(\frac{n}{360})$$: $$\frac{60}{360} = \frac{1}{6}$$

Step 2: Calculate the full area value of the pie $$(\pi \times r^2)$$: $$3.14 \times 6 \times 6 = 113.04$$ square cm

Step 3: Multiply the fraction by the total area (SecA): $$\frac{1}{6} \times 113.04 = 18.84$$ square cm

Practice problems

Problem 7
A circular garden patch has a total area of 72 square feet. A shaded section of this garden forms a sector with a central angle of 45 degrees. Find the area of this shaded sector.

Show solution

Answer: 9 square feet

Step 1: Convert the degree angle into a simplified fraction $$(\frac{n}{360})$$: $$\frac{45}{360} = \frac{1}{8}$$
Step 2: Identify the given total circle area directly $$(\pi \times r^2)$$: 72
Step 3: Calculate the area of the sector segment (SecA): $$\frac{1}{8} \times 72 = 9$$ square feet
Problem 8
A circular spinner with a radius of 4 inches is divided into four equal quadrants. One of the quadrants is shaded gray. What is the area of the shaded quadrant?

Show solution

Answer: 12.56 square inches

Step 1: Recognize that one equal quadrant represents a central angle (n) of 90 degrees, which simplifies to $$\frac{1}{4}$$ of the circle.
Step 2: Compute the full area value of the circle: $$3.14 \times 4 \times 4 = 50.24$$
Step 3: Multiply by the quadrant fraction (SecA): $$\frac{1}{4} \times 50.24 = 12.56$$ square inches
Problem 9
An engineer designs a circular dial plate where a semi-circular sector (180 degrees) is shaded blue. If the radius of the dial plate is 10 cm, find the area of the blue shaded region.

Show solution

Answer: 157 square cm

Step 1: Set up the area fraction ratio $$(\frac{n}{360})$$: $$\frac{180}{360} = \frac{1}{2}$$ (half a circle)
Step 2: Calculate the full area parameter of the circle: $$3.14 \times 10 \times 10 = 314$$
Step 3: Multiply the fraction by the total area (SecA): $$0.5 \times 314 = 157$$ square cm

Remember

Simplify the angle fraction $$(\frac{n}{360})$$ first (e.g., $$\frac{90}{360}$$ reduces cleanly to $$\frac{1}{4}$$) to make multiplication straightforward.