What is the Area Bounded by Composite Regions?

S7-SA1-0309

Grade Level:

Class 12

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Definition

What is it?

The area bounded by composite regions refers to finding the total area of shapes that are made up of two or more simpler, basic geometric shapes like squares, rectangles, triangles, or circles. Instead of a single shape, you combine areas from different parts to get the total.

Simple Example

Quick Example

Imagine you are painting a wall in your room. This wall has a rectangular main section and a triangular decorative top. To buy enough paint, you need to know the total area of the wall. You would find the area of the rectangle and the area of the triangle separately, then add them up to get the 'composite area'.

Worked Example

Step-by-Step

PROBLEM: Find the area of a garden shaped like a rectangle joined with a semi-circle. The rectangle has a length of 10 meters and a width of 7 meters. The semi-circle is attached to the 7-meter side of the rectangle.

STEP 1: Identify the individual shapes. We have a rectangle and a semi-circle.
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STEP 2: Calculate the area of the rectangle. Area_rectangle = length * width = 10 m * 7 m = 70 square meters.
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STEP 3: Determine the dimensions of the semi-circle. The semi-circle is attached to the 7-meter side, so its diameter is 7 meters. The radius (r) will be diameter / 2 = 7 / 2 = 3.5 meters.
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STEP 4: Calculate the area of the full circle. Area_circle = pi * r^2. Using pi = 22/7, Area_circle = (22/7) * (3.5)^2 = (22/7) * 12.25 = 22 * 1.75 = 38.5 square meters.
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STEP 5: Calculate the area of the semi-circle. Area_semi-circle = Area_circle / 2 = 38.5 / 2 = 19.25 square meters.
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STEP 6: Add the areas of the individual shapes to get the total composite area. Total Area = Area_rectangle + Area_semi-circle = 70 + 19.25 = 89.25 square meters.

ANSWER: The total area of the garden is 89.25 square meters.

Why It Matters

Common Mistakes

MISTAKE: Forgetting to break down the complex shape into simpler parts | CORRECTION: Always start by visually dividing the composite region into basic shapes whose area formulas you already know.

MISTAKE: Using the wrong dimensions for a part, especially when a side is shared. For example, using the length of a rectangle as the radius of a semi-circle directly. | CORRECTION: Carefully identify if a shared side is a diameter, radius, or a side of a rectangle for each individual shape.

MISTAKE: Mixing up addition and subtraction of areas. Sometimes you need to subtract an 'empty' region. | CORRECTION: Visualize if the smaller shapes are 'filling' the larger shape (add) or if a part is 'cut out' from a larger shape (subtract).

Practice Questions

Try It Yourself

QUESTION: A school playground is shaped like a square with side 20 meters, and a triangle attached to one side. The triangle has a base of 20 meters and a height of 8 meters. What is the total area of the playground? | ANSWER: 480 square meters

QUESTION: A window pane is a rectangle 1.5 meters long and 0.8 meters wide, with a small circular hole of radius 0.1 meters cut out from its center. What is the area of the glass remaining? (Use pi = 3.14) | ANSWER: 1.1686 square meters

QUESTION: A design on a rangoli is made of two identical semi-circles joined to opposite sides of a square. The square has a side of 14 cm. Find the total area of the rangoli design. (Use pi = 22/7) | ANSWER: 308 square cm

MCQ

Quick Quiz

A park has a rectangular section 20m by 15m, and a triangular section attached to the 20m side with a height of 5m. What is the total area of the park?

300 square meters

350 square meters

400 square meters

450 square meters

The Correct Answer Is:

The area of the rectangle is 20 * 15 = 300 sq m. The area of the triangle is (1/2) * base * height = (1/2) * 20 * 5 = 50 sq m. The total area is 300 + 50 = 350 sq m.

Real World Connection

In the Real World

When city planners design new infrastructure, like a metro station or a flyover, they often deal with land parcels that are not perfect squares or circles. They use the concept of composite areas to calculate the exact land needed, estimate construction materials, or even design efficient traffic flow paths, much like how ISRO engineers calculate surface areas for spacecraft components.

Key Vocabulary

Key Terms

COMPOSITE REGION: A shape made up of two or more simpler shapes. | AREA: The amount of surface covered by a 2D shape. | DIMENSIONS: Measurements like length, width, height, or radius. | GEOMETRIC SHAPES: Basic shapes like squares, circles, triangles. | SEMI-CIRCLE: Half of a circle.

What's Next

What to Learn Next

Great job learning about composite areas! Next, you can explore 'Area Between Curves Using Integration'. This will teach you how to find areas of even more complex shapes using advanced math tools, which is super useful in higher studies and competitive exams!