AREA
AREA — Study Notes
NCERT-aligned · 9 notes · 3 shown free
Introduction
ExplanationIntroduction
In this chapter, we explore two important types of quadrilaterals: rectangles and squares. These shapes are fundamental in geometry and have many practical applications in daily life, architecture, and design. A rectangle is a quadrilateral with four right angles, and opposite sides equal and parallel. A square is a special type of rectangle where all four sides are equal in length, and all angles are right angles. Understanding these shapes helps in learning about area, perimeter, and properties of quadrilaterals. The chapter begins by defining these shapes and then moves on to their properties, formulas for area and perimeter, and real-life examples. The study of rectangles and squares also introduces the concept of diagonals and their properties, which is essential for understanding symmetry and congruence in geometry. This chapter builds on earlier knowledge of polygons and quadrilaterals and prepares students for more advanced topics involving shapes and measurements.
- Rectangle: quadrilateral with four right angles and opposite sides equal
- Square: rectangle with all sides equal
- Both shapes are fundamental in geometry and have practical applications
- Understanding area and perimeter formulas is key
- Diagonals play an important role in the properties of these shapes
- Builds on previous knowledge of polygons and quadrilaterals
- 📌 Rectangle: A quadrilateral with four right angles and opposite sides equal
- 📌 Square: A rectangle with all sides equal in length
Properties of Rectangle
ExplanationProperties of Rectangle
A rectangle is a quadrilateral with four right angles (each 90°). The opposite sides of a rectangle are equal and parallel. This means that if one side is of length 'l' and the adjacent side is 'b', then the opposite sides will also be 'l' and 'b' respectively. The diagonals of a rectangle are equal in length and bisect each other. This property is useful in proving congruence and similarity in geometry. The sum of the interior angles of a rectangle is 360°, as with any quadrilateral. Rectangles are widely used in construction and design due to their regular shape and symmetry. Understanding these properties helps in calculating area, perimeter, and other measurements accurately. The rectangle is a special type of parallelogram where all angles are right angles. This section also explains how to identify rectangles in different orientations and how their properties remain consistent regardless of rotation or reflection.
- All four angles are right angles (90°)
- Opposite sides are equal and parallel
- Diagonals are equal and bisect each other
- Sum of interior angles is 360°
- Rectangle is a special parallelogram with right angles
- Properties hold true regardless of orientation
- 📌 Diagonal: A line segment joining two opposite vertices of a polygon
- 📌 Bisect: To divide into two equal parts
Properties of Square
ExplanationProperties of Square
A square is a special type of rectangle where all four sides are equal in length, and all four angles are right angles (90°). Because a square is a rectangle, it inherits all properties of a rectangle, such as opposite sides being parallel and diagon
Practice Questions — AREA
Includes NCERT exercise questions with answers
Q1.4. [Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.
Answer:
To convert an isosceles trapezium to a rectangle of equal area using dissection, one can cut the trapezium into parts and rearrange them to form a rectangle. The method involves drawing a perpendicular from one of the non-parallel sides to the base, then cutting along this line and rearranging the parts so that the height remains the same and the base becomes the average of the two parallel sides, thus forming a rectangle with the same area as the trapezium.
Explanation:
The area of a trapezium is given by 1/2 × height × (sum of parallel sides). By dissecting the trapezium and rearranging the parts, we form a rectangle whose length is the average of the two parallel sides and height is the same as the trapezium's height, ensuring equal area.
Q2.5. Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area— Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH? [Hint: If Δ AHI ≅ Δ DGI and Δ BEJ ≅ Δ CFJ, then the trapezium and rectangle have equal areas.]
Answer:
To find the vertices of rectangle EFGH corresponding to trapezium ABCD, we use the given congruences: 1. Identify points I and J on the trapezium such that triangles AHI and DGI are congruent, and triangles BEJ and CFJ are congruent. 2. Using these congruent triangles, construct rectangle EFGH such that its sides correspond to the height and the average length of the parallel sides of trapezium ABCD. 3. The rectangle EFGH will have the same area as trapezium ABCD because the congruent triangles ensure that the area lost or gained in rearrangement is balanced. Thus, vertices E, F, G, and H are determined by the points of intersection and congruent triangles ensuring equal area.
Explanation:
The method relies on dissecting trapezium ABCD into parts and rearranging them to form rectangle EFGH. The congruent triangles ensure that the rearrangement does not change the total area. The height of the rectangle equals the height of the trapezium, and the length equals the average of the two parallel sides.
Q3.6. Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area 144 cm².
Answer:
To construct a trapezium of area 144 cm²: 1. Decide the height (h) of the trapezium. 2. Choose lengths of the two parallel sides (a and b) such that the area formula holds: Area = 1/2 × h × (a + b) = 144 cm² 3. For example, if h = 12 cm, then (a + b) = (2 × 144) / 12 = 24 cm. 4. Choose a = 10 cm and b = 14 cm (sum 24 cm). 5. Draw trapezium with these dimensions. This trapezium will have area 144 cm².
Explanation:
Using the area formula for trapezium, we select height and sum of parallel sides to satisfy the given area. Then, by drawing the trapezium with these dimensions, the required figure is constructed.
Q4.7. A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.
Answer:
Let the side of the regular hexagon be 's'. 1. The regular hexagon can be divided into 6 equilateral triangles of side s. 2. The given trapezium, equilateral triangle, and rhombus are parts of these triangles. 3. Calculate the area of each shape using known formulas: - Area of equilateral triangle = (√3 / 4) × s² - Area of rhombus = (1/2) × product of diagonals (which can be expressed in terms of s) - Area of trapezium = (1/2) × height × sum of parallel sides (expressed in terms of s) 4. Compute each area and simplify the ratio. The exact ratio depends on the dimensions given in the figure, but the method involves expressing all areas in terms of s² and then simplifying.
Explanation:
By expressing all areas in terms of the side length and using area formulas for each shape, the ratio of areas can be found by dividing each area by the others.
Q5.8. ZYXW is a trapezium with ZY || WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of Δ ZWB.
Answer:
Given trapezium ZYXW with ZY parallel to WX and A as midpoint of XY. 1. Join points Z and W. 2. Since A is midpoint of XY, line ZA and AW can be used to relate areas. 3. By constructing triangle ZWB appropriately and using properties of trapezium and midpoint, prove that area of trapezium ZYXW equals area of triangle ZWB. This can be shown by comparing the base and height of both shapes or by using coordinate geometry or area subtraction methods.
Explanation:
The proof involves showing that the trapezium and the triangle share the same base and height or can be decomposed into equal areas, thus proving their areas are equal.
Q6.What do you think is the area of an A4 sheet? Its sidelengths are 21 cm and 29.7 cm. Now find its area.
Answer:
Area of A4 sheet = length × breadth = 29.7 cm × 21 cm = 623.7 cm².
Explanation:
Area of rectangle = length × breadth. Multiplying given dimensions gives the area.
Q7.What do you think is the area of the tabletop that you use at school or at home? You could perhaps try to visualise how many A4 sheets can fit on your table.
Answer:
This is an open-ended question. Measure or estimate the dimensions of your tabletop, calculate its area by multiplying length and breadth, then divide by the area of one A4 sheet (623.7 cm²) to estimate how many A4 sheets fit on the table.
Explanation:
Area of tabletop = length × breadth. Number of A4 sheets = (Area of tabletop) / (Area of one A4 sheet).
Q8.Express the following lengths in centimeters: (i) 5 in (ii) 7.4 in
Answer:
(i) 5 in = 5 × 2.54 cm = 12.7 cm (ii) 7.4 in = 7.4 × 2.54 cm = 18.796 cm
Explanation:
1 inch = 2.54 cm. Multiply given inches by 2.54 to convert to centimeters.
All 7 Chapters in Ganita Prakash Part-II
Mathematics · Class 8