Mensuration: Perimeter, Area, and Volume

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Questions and Answers

A pentagon ABCDE is divided into three parts by constructing two diagonals. Which of the following equations accurately represents the total area of the pentagon?

  • area ABCDE = area of ∆ABD + area of ∆ACE + area of ∆ADE
  • area ABCDE = area of ∆ABC + area of ∆ACD + area of ∆AED (correct)
  • area ABCDE = area of ∆ABC + area of ∆ACD - area of ∆AED
  • area ABCDE = area of ∆ABC - area of ∆ACD + area of ∆AED

A pentagon ABCDE is divided using one diagonal AD and two perpendiculars BF and CG. Which areas would you sum to find the total area of the pentagon?

  • area of right-angled AFB + area of trapezium BFGC - area of right-angled CGD + area of AED
  • area of right-angled AFB + area of trapezium BFGC + area of right-angled CGD + area of AED (correct)
  • area of AFB + area of BFGC + area of CGD - area of AED
  • area of right-angled AFB - area of trapezium BFGC + area of right-angled CGD + area of AED

Polygon ABCDE is divided into several parts. Given AD = 8 cm, AH = 6 cm, AG = 4 cm, AF = 3 cm, BF = 2 cm, CH = 3 cm, and EG = 2.5 cm, which calculation correctly represents the area of trapezium FBCH?

  • $( \frac{5}{2})$
  • $(3 \times \frac{5}{2})$ (correct)
  • $(5 \times \frac{5}{2})$
  • $(2 \times \frac{5}{2})$

The area of a trapezium-shaped field is 480 $m^2$. The distance between two parallel sides is 15 m, and one of the parallel sides is 20 m. What is the length of the other parallel side?

<p>44 m (C)</p> Signup and view all the answers

A rhombus has an area of 240 $cm^2$, and one of its diagonals measures 16 cm. What is the length of the other diagonal?

<p>30 cm (C)</p> Signup and view all the answers

A hexagon MNOPQR with sides of 5 cm is divided into a rectangle and two congruent triangles. If the height of each triangle is 3 cm and the base of the rectangle is 8 cm, what is the area of the hexagon?

<p>64 $cm^2$ (D)</p> Signup and view all the answers

A table top is shaped like a trapezium with parallel sides of 1 m and 1.2 m and a perpendicular distance of 0.8 m between them. What is the area of the table top?

<p>0.88 $m^2$ (C)</p> Signup and view all the answers

The area of a trapezium is 34 $cm^2$, and the length of one of the parallel sides is 10 cm, with a height of 4 cm. What is the length of the other parallel side?

<p>7 cm (C)</p> Signup and view all the answers

The diagonal of a quadrilateral-shaped field is 24 m, and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. Find the area of the field.

<p>252 $m^2$ (A)</p> Signup and view all the answers

The floor of a building consists of 3000 tiles, each shaped like a rhombus with diagonals of 45 cm and 30 cm. If the cost of polishing is ₹4 per $m^2$, what is the total cost of polishing the floor?

<p>₹2700 (A)</p> Signup and view all the answers

A suitcase measures 80 cm x 48 cm x 24 cm. It is to be covered with tarpaulin cloth. What length of tarpaulin of width 96 cm is required to cover 100 such suitcases?

<p>192 m (B)</p> Signup and view all the answers

Rukhsar painted the outside of a cabinet measuring 1 m x 2 m x 1.5 m. If she painted all except the bottom of the cabinet, how much surface area did she cover?

<p>11 $m^2$ (B)</p> Signup and view all the answers

Daniel is painting the walls and ceiling of a cuboidal hall with dimensions 15 m x 10 m x 7 m. If each can of paint covers 100 $m^2$, how many cans of paint will he need to paint the room?

<p>4 (B)</p> Signup and view all the answers

A company packages its milk powder in a cylindrical container with a base diameter of 14 cm and a height of 20 cm. If the company places a label around the surface, 2 cm from the top and bottom, what is the area of the label?

<p>1056 $cm^2$ (B)</p> Signup and view all the answers

What is the volume of a cube with a side of 4 cm?

<p>64 $cm^3$ (A)</p> Signup and view all the answers

A cuboid has dimensions of 8 cm x 3 cm x 2 cm. What is its volume?

<p>48 $cm^3$ (B)</p> Signup and view all the answers

What is the formula to calculate the total surface area of a cuboid?

<p>$2(lb+bh+hl)$ (B)</p> Signup and view all the answers

Two cubes, each with side length 'b', are joined to form a cuboid. What is the surface area of the resulting cuboid?

<p>$10b^2$ (A)</p> Signup and view all the answers

A godown is in the form of a cuboid measuring 60 m x 40 m x 30 m. How many cuboidal boxes can be stored in it if the volume of one box is 0.8 $m^3$?

<p>90000 (A)</p> Signup and view all the answers

A rectangular piece of paper with a width of 14 cm is rolled along its width to form a cylinder. If the radius of the cylinder is 20 cm, what is the volume of the formed cylinder?

<p>17600 $cm^3$ (A)</p> Signup and view all the answers

Flashcards

Perimeter

The distance around a closed plane figure's boundary.

Area

The region covered by a closed plane figure.

Area of polygon

A method to find the area of polygons by dividing them into triangles and other shapes.

Volume

The amount of space occupied by a three-dimensional object.

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Capacity

The quantity that a container can hold.

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Right Circular Cylinder

A cylinder where the line segment joining the centers of circular faces is perpendicular to the base.

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Surface area of a solid

The sum of the areas of all its faces.

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Surface area of a cuboid

2(lb + bh + hl), where l = length, b = breadth, h = height.

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Surface area of a cube

6l², where l is the length of a side.

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Surface area of a cylinder

2Ï€r(r + h), where r = radius, h = height

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Volume of a Cuboid

l × b × h

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Volume of a Cube

l³ (l cubed)

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Volume of a Cylinder

πr²h (pi * radius squared * height)

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Study Notes

Introduction to Mensuration

  • Perimeter is the distance around the boundary of a closed plane figure
  • Area is the region covered by a closed plane figure
  • This chapter solves problems related to the perimeter and area of quadrilaterals, and the surface area and volume of solids like cubes, cuboids, and cylinders

Area of a Polygon

  • Quadrilaterals can be split into triangles to calculate their area

Area of a Pentagon (Using Diagonals)

  • By constructing two diagonals, a pentagon can be divided into three triangles
  • The area of the pentagon is the sum of the areas of the three triangles

Area of a Pentagon (Using a Diagonal and Perpendiculars)

  • By constructing one diagonal and two perpendiculars on it, a pentagon can be divided into four parts: two right-angled triangles and a trapezium
  • The area of the pentagon equals the sum of the areas of the right-angled triangles, and the trapezium

Dividing Polygons for Area Calculation

  • Polygons can be divided into triangles and trapeziums to ascertain their area

Area of a Trapezium - Example

  • Area of a trapezium-shaped field is 480 m² with the distance between parallel sides being 15 m
  • If one of the parallel sides is 20 m, then the other parallel side is determined to be 44 m

Area of a Rhombus - Example

  • The area of a rhombus is 240 cm², with one diagonal being 16 cm
  • Therefore, the length of the second diagonal is 30 cm

Area of a Hexagon - Example

  • A hexagon (MNOPQR) is divided into two congruent trapeziums by NQ
  • If the sides are 5 cm, the area of each trapezium (MNQR) is 32 cm²
  • The area of the hexagon MNOPQR is 64 cm²

Finding area using an alternative method

  • A hexagon shape can be split into 2 congruent triangles and a rectangle
  • This gives an area of 64cm²

Exercise 9.1 - Area of a Trapezium

  • The top surface of a table is shaped like a trapezium
  • Parallel sides are 1 m and 1.2 m, and the perpendicular distance between them is 0.8 m
  • The area of the top surface can be determined using these values

Finding Total Cost

  • Polishing a floor consisting of 3000 rhombus-shaped tiles, each with diagonals of 45 cm and 30 cm, is considered
  • The total cost of polishing, given a cost of ₹4 per m², is calculated

Surface Area

  • 2-D figures form the faces of 3-D figures

Cuboid

  • All six faces are rectangular, and opposite faces are identical (three pairs of identical faces)
  • The surface area of a cuboid is the sum of the areas of all its faces

Cylinder Properties

  • Cylinders have congruent circular faces parallel to each other
  • A line segment joining the centers of circular faces is perpendicular to the base
  • These cylinders are right circular cylinders

Surface Area Calculation

  • The total surface area can be found by adding the area of each face

Total Surface Area of a Cuboid

  • The total surface area of a cuboid is the sum of the areas of all its faces: 2(h × l) + 2(b × l) + 2(b × h) which simplifies to 2(lb + bh + hl)

Lateral Surface Area of a Cuboid

  • Side walls (excluding the top and bottom) make up the lateral surface area of the cuboid
  • The lateral surface area of a cuboid is given by 2(h × l) + 2(b × h) or 2h(l + b)

Relationship Between Total and Lateral Surface Area of a Cuboid

  • The total surface area of a cuboid equals the lateral surface area plus 2 times the area of its base

Cube Properties

  • All faces of a cube are square in shape, making length, height, and width equal

Total Surface Area of a Cube

  • The total surface area of a cube is 6l², where l is the length of a side

Curved Surface Area of a Cylinder

  • The paper that fits around the can is rectangular
  • The area of the rectangular strip is 2Ï€rh

Total Surface Area of a Cylinder deduced from dissection

  • The curved surface constitutes 2Ï€rh and there are two circular faces
  • Total Surface Area is 2Ï€r² + 2Ï€rh

Determining the Area of Paper Needed for an Aquarium

  • An aquarium is in the form of a cuboid, with external measures of length l = 80 cm, width b = 30 cm, and height h = 40 cm
  • The area of the base is l × b = 2400 cm², the area of the side face is b × h = 1200 cm², and the area of the back face is l × h = 3200 cm²
  • paper covers the base, back face, and two side faces is calculated to be 8000 cm²

Internal Measures Example

  • Find the cost of whitewashing, including the ceiling requires an additional calculation of the ceiling area and its respective cost

Cylindrical Pillars Example

  • Radii and height are needed to calculate total cost of painting. The total can be scaled up based on pillar counts

Volume Definition

  • Amount of space an object occupies

Volume Measurement

  • Cubic units are used to measure the volume

Volume of Cuboid

  • l × b × h

Relationship

Area of its base × height

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