Area of Composite Figures

In geometry, a compound figure or mixed shape is a figure made up of two or more simple geometric shapes such as rectangles, squares, triangles, circles, and semicircles. Calculating the area of compound shapes involves finding the areas of the simple shapes and then adding them together or, in some cases, subtracting them. This concept can be compared to solving a puzzle, where the pieces are individual shapes that fit together to form a bigger picture.

Division of composite data

When dealing with compound shapes, the main thing is to break down the complex shape into smaller, more manageable parts. These parts should be shapes whose area we can easily calculate using known formulas.

Basic shapes and their area formulas

• Rectangle: Area = length × width
• Square: Area = side × side
• Triangle: Area = (base × height) / 2
• Circle: Area = π × radius²
• Semicircle: Area = (π × radius²) / 2

For example, if a compound figure contains a rectangle and a triangle, we would use the area of a rectangle and the area of a triangle formulas, and then sum them to find the total area of the compound figure.

Visual example: mixed shape with a rectangle and a triangle

To find the area of this composite figure, first find the area of the rectangle and the area of the triangle:

Rectangle area = 100 (length) × 50 (width) = 500 square units
Area of the triangle = 0.5 × 50 (base) × 50 (height) = 125 square units
Total area = rectangle area + triangle area = 500 + 125 = 625 square units

Text example 1

Imagine you have a garden that's shaped like a large L, which means it can be divided into a large rectangle and a small square. The larger rectangular part of the L is 8 metres long and 3 metres wide, while the smaller square is 3 metres on each side.

Area of the rectangle = 8 × 3 = 24 square meters
Square area = 3 × 3 = 9 square meters
Total garden area = 24 + 9 = 33 sq.m

Visual example: mixed shape with a rectangle and a semicircle

Consider finding the area of such a composite figure. First, calculate the areas using their respective formulas and then add them:

Rectangle area = 180 (length) × 60 (width) = 10,800 square units
Semicircle area = (π × 90²) / 2 = (π × 8100) / 2 ≈ 12,733 / 2 ≈ 6,367 square units
Total area = rectangle area + semicircle area = 10,800 + 6,367 ≈ 17,167 square units

Text example 2

Suppose you want to carpet the floor of your room, which consists of a rectangular area and a semicircular area joined together. The rectangle is 12 m by 5 m, and the radius of the semicircle is 5 m.

Area of the rectangle = 12 × 5 = 60 square meters
Area of semicircle = (π × 5²) / 2 ≈ (3.14159265 × 25) / 2 ≈ 39.27 m²
Total carpet area ≈ 60 + 39.27 ≈ 99.27 sq.m

Mixed numbers involving subtraction

Sometimes, compound shapes need to be subtracted if parts of one simple shape overlap another. When parts overlap, calculate the area of each simple shape and subtract the overlapping area to find the actual area of the compound shape.

Text example 3

Imagine a picture frame which is a rectangle of external dimensions 10 cm by 8 cm and has a circular hole of radius 2 cm cut through its centre.

Area of the outer rectangle = 10 × 8 = 80 sq. cm
Circle hole area = π × 2² = 3.14159265 × 4 ≈ 12.57 sq. cm
Frame area = outer rectangle area - circle hole area = 80 - 12.57 ≈ 67.43 sq. cm

Visual example: compound shape with circle inside rectangle (subtraction)

In this example, calculate the total area by subtracting the area of the circle from the area of the rectangle.

Practical applications of composite shapes

Understanding how to find the area of compound shapes can be very useful in real-life situations. It can be applied in renovating houses, landscaping gardens, designing rooms, and even constructing buildings. This knowledge helps in accurately estimating the requirements of materials such as paint, flooring, or tiles and helps in estimating costs in such projects.

The idea is to always break down any unfamiliar figure into familiar components. By breaking them down, you can perform the calculations in a systematic way and arrive at a solution with confidence.

Practice problems

Try calculating the area of these composite shapes:

• A park that is in the shape of a rectangle and adjoins a semicircle. The rectangle measures 20 m by 10 m, and the diameter of the semicircle is equal to the smaller side of the rectangle.
• An L-shaped swimming pool, with one side measuring 15 ft by 8 ft and the other side (attached) being a square measuring 8 ft on each side.
• An island with a rectangular centre of dimensions 60 km by 40 km, surrounded by a circular lagoon with an outer radius of 50 km which completely encloses the rectangular area.

Conclusion

Understanding and calculating the area of compound shapes may seem challenging at first, but by breaking down the shapes into simpler shapes and applying familiar area formulas, it becomes manageable. Remember to carefully add and subtract the required values depending on whether the shapes overlap or are combined separately. Keep practicing different configurations, and soon, you will be able to solve such problems quickly and accurately.

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