Areas of Combinations of Plane Figures

Understanding the areas of combinations of plane figures involves calculating the area of shapes formed by combining two or more basic geometric figures like squares, rectangles, triangles, circles, and parallelograms. This often requires breaking down complex shapes into simpler, known shapes, calculating the area of each simpler shape, and then combining those areas using addition or subtraction (depending on how the shapes are combined).

The core concept relies on these fundamental principles:

• Decomposition: Breaking down a complex shape into simpler, manageable components.
• Area Formulas: Knowing and applying the standard area formulas for basic shapes.
• Addition/Subtraction: Using addition to find the total area of combined shapes and subtraction to find the area of a shape with a cutout or overlapping parts.

Formulae

Here are some essential area formulas you’ll need:

• Rectangle: $Area = length \times width$
• Square: $Area = side \times side = side^2$
• Triangle: $Area = \frac{1}{2} \times base \times height$
• Circle: $Area = \pi \times radius^2$ (where $\pi \approx 3.14159$)
• Parallelogram: $Area = base \times height$
• Trapezoid: $Area = \frac{1}{2} \times (base_1 + base_2) \times height$

Examples

Example-1: A rectangular garden is 10 meters long and 6 meters wide. A circular fountain with a radius of 2 meters is placed in the garden. What is the area of the garden not occupied by the fountain?

Solution:

1. Calculate the area of the rectangle: $Area_{rectangle} = length \times width = 10\,m \times 6\,m = 60\,m^2$
2. Calculate the area of the circle: $Area_{circle} = \pi \times radius^2 = \pi \times (2\,m)^2 = 4\pi\,m^2 \approx 12.57\,m^2$
3. Subtract the area of the circle from the area of the rectangle: $Area_{garden} = Area_{rectangle} – Area_{circle} = 60\,m^2 – 12.57\,m^2 \approx 47.43\,m^2$

The area of the garden not occupied by the fountain is approximately $47.43\,m^2$.

Example-2: A composite shape is formed by a square with side 8cm and a triangle on top of it with base 8cm and height 5cm. What is the total area of the composite shape?

Solution:

1. Calculate the area of the square: $Area_{square} = side^2 = (8\,cm)^2 = 64\,cm^2$
2. Calculate the area of the triangle: $Area_{triangle} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 8\,cm \times 5\,cm = 20\,cm^2$
3. Add the areas of the square and the triangle: $Area_{total} = Area_{square} + Area_{triangle} = 64\,cm^2 + 20\,cm^2 = 84\,cm^2$

The total area of the composite shape is $84\,cm^2$.

Common mistakes by students

• Incorrectly Identifying Shapes: Misidentifying the shapes that make up the composite figure (e.g., confusing a parallelogram with a rectangle).
• Using Wrong Formulas: Applying the wrong area formula for a given shape.
• Incorrect Units: Forgetting to include units (e.g., cm², m²) or using incorrect units.
• Forgetting to Subtract Overlapping Areas: When a shape has an overlap, the overlapping area needs to be subtracted. Students sometimes forget to do this.
• Using Incorrect Measurements: Using wrong measurements of the sides or radius.

Real Life Application

The concepts of areas of combinations of plane figures are widely used in real-life scenarios:

• Construction: Calculating the amount of paint needed to paint a wall with windows and doors, or determining the amount of flooring needed for a room.
• Architecture: Designing and calculating the area of floor plans, roofs, and facades of buildings.
• Landscaping: Determining the amount of grass seed needed for a lawn with flowerbeds, or calculating the area of a garden with different zones.
• Manufacturing: Calculating the area of materials needed to produce a product, such as sheet metal or fabric.
• Interior Design: Calculating the area of a room for furniture placement and carpet selection.

Fun Fact

The problem of calculating areas has been around for thousands of years. Ancient civilizations like the Egyptians and Babylonians developed methods for approximating areas, even before the sophisticated formulas we use today. They used these methods for practical purposes like land surveying and taxation.

Recommended YouTube Videos for Deeper Understanding

Q.1 A rectangular garden is 20 meters long and 15 meters wide. A path of uniform width 2 meters is built around the outside of the garden. What is the area of the path?

Check Solution

Ans: A

The outer dimensions of the path are (20 + 2*2) meters and (15 + 2*2) meters, or 24 meters and 19 meters. The area of the path is the difference between the area of the outer rectangle and the area of the garden: $24*19 – 20*15 = 456 – 300 = 156$.

Q.2 A square is inscribed in a circle of radius 5 cm. What is the area of the region inside the circle but outside the square?

Check Solution

Ans: A

The diagonal of the square is the diameter of the circle, which is $2*5 = 10$ cm. If the side length of the square is $s$, then $s^2 + s^2 = 10^2$, so $2s^2 = 100$ and $s^2 = 50$. Thus the area of the square is 50 cm$^2$. The area of the circle is $\pi r^2 = \pi(5^2) = 25\pi$ cm$^2$. The area of the region inside the circle but outside the square is $25\pi – 50$.

Q.3 A composite figure consists of a rectangle with length 10 cm and width 6 cm, and a semicircle attached to one of the longer sides. What is the area of the composite figure?

Check Solution

Ans: C

The area of the rectangle is $10 * 6 = 60$ cm$^2$. The semicircle has a diameter of 10 cm, so its radius is 5 cm. The area of the semicircle is $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi (5^2) = \frac{25\pi}{2} = 12.5\pi$ cm$^2$. The area of the composite figure is $60 + 12.5\pi$. Not an option

Q.4 Two circles with radii 3 cm and 5 cm are externally tangent. What is the area of the region between the circles?

Check Solution

Ans: A

The area of the larger circle is $\pi (5^2) = 25\pi \, cm^2$. The area of the smaller circle is $\pi (3^2) = 9\pi \, cm^2$. The area of the region between the circles is the difference: $25\pi – 9\pi = 16\pi \, cm^2$.

Q.5 A right triangle has legs of length 6 cm and 8 cm. A square is constructed on the hypotenuse of the triangle. What is the area of the square?

Check Solution

Ans: C

The length of the hypotenuse is $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm. The area of the square is the side length squared, which is $10^2 = 100$ cm$^2$.

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