Category: Figures and Volumes

Introduction

In this lesson, we solve word problems involving the volume of a triangular prism. Real world problems.

Example 1

A right prism base is a triangle whose sides are 18 cm, 20 cm and 34 cm. Find the volume of the prism if its height is 9 cm.

Solution

Step 1:

Area of baseA = s(s−a)(s−b)(s−c)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√

= 36(36−18)(36−20)(36−34)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√

= 144 square cm;

height h = 9 cm

Step 2:

Volume of triangular prism V = Ah = 144(9)

= 1296 cu cm

Example 2

The base of a right prism is a right triangle with legs 12 and 5 cm. If the volume of the prism be 390 cu cm, find the height of the prism.

Solution

Step 1:

Area base A = 12 × 12 × 5 = 30 sq cm;

Volume V = 390 cu cm

Step 2:

Height of triangular prism h = V/A = 390/30

= 13 cm

Introduction

In this lesson we find the volume of a triangular prism

A triangular prism is a prism that has two congruent parallel triangles as its bases and rectangular lateral faces.

Formula for the volume of a triangular prism

If A is the area of the base triangle and h is the height of the prism then volume of the prism is given by

Volume V = A × h

Where A = 12 b h or s(s−a)(s−b)(s−c)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√ or a23‾√/4

b is the base of the triangle and h is the height

a, b, and c are the sides of the triangle and s =(a+b+c)/2

a is the side of an equilateral triangle

Example 1

Find the volume of the following triangular prism.

Solution

Step 1:

Volume of a triangular prism = Areas of base triangle × height of prism

Step 2:

Volume of given prism V = 12 × 14 × 8 × 10

= 560 cubic feet

Example 2

Find the volume of the following triangular prism.

Solution

Step 1:

Volume of a triangular prism = Areas of base triangle × height of prism

Step 2:

Volume of given prism V = 12 × 14 × 8 × 6

= 336 cubic ft

Introduction

In this lesson we solve word problems involving the rate of filling or emptying a rectangular prism.

Formula to find time to drain or fill up a rectangular prism at a certain rate.

If the rate of filling up or draining a rectangular prism is r units per minute and if the volume of the prism is V cubic units, then the time taken is given by

Time taken to drain/fill up = Volume/ rate = V/r

Example 1

Stacy is cleaning out her fish tank that is 6 feet long, 4 feet wide, and 3 feet deep. It is 70% full of water and drains at the rate of 2 cubic foot per minute. How long does it take to drain completely?

Solution

Step 1:

Volume of water = 0.7 × 6ft × 4ft × 3ft

= 50.4 cu ft

Step 2:

Time taken to drain = Volume/rate = 50.4/2

= 25.2 minutes

Example 2

A rectangular water tank 8 m × 1.8 m × 3.5 m is 1/5 full of water. How long does it take to fill the tank completely, if it fills up at the rate of 2 kilolitres per minute?

Solution

Step 1:

Volume of water to fill = 4/5 × 8.0 m × 1.8 m × 3.5 m

= 40.32 cu m

Step 2:

Time taken to fill up = Volume/rate = 40.32/2

= 20.16 minutes

Introduction

In this lesson we solve word problems involving the volume of a rectangular prism.

Example 1

Sarah has a chocolate box whose length is 12 cm, height 9 cm, and width 6 cm. Find the volume of the box.

Solution

Step 1:

The given box has length = 12 cm; width = 6 cm; height = 9 cm.

Step 2:

The volume of box V = l × w × h = 12 × 6 × 9

= 648 cubic cm

Example 2

A water tank is 90 m long and 60 m wide. What is the volume of the water in the tank, if the depth of water is 40 m?

Solution

Step 1:

The given tank has length = 90 m; width = 60 m; height = 40 m.

Step 2:

The volume of water tank V = l × w × h = 90 × 60 × 40

= 216000 cubic m.

Introduction

In this lesson, we find the volume of rectangular prisms with fractional edge lengths.

Formula for the volume of solid made of cubes with unit fractional edge lengths

l = number of cubes with unit fractional edge length along the length

w = number of cubes with unit fractional edge length along the width

h = number of cubes with unit fractional edge length along the height

k = unit fractional edge length

Volume of solid = l × k × w × k × h × k cubic units

Example 1

Find the volume of following solid of cubes with unit fraction edge lengths. Each prisms unit is measured in cm (not to scale)

Solution

Step 1:

Solid of cubes with unit fraction edge lengths

Step 2:

Volume V = l w h = 634×3×4

= 9×34×4×34×163×34

= 81 cu cm

Example 2

Find the volume of following solid of cubes with unit fraction edge lengths. Each prisms unit is measured in cm (not to scale)

Solution

Step 1:

Solid of cubes with unit fraction edge lengths

Step 2:

Volume V = l w h = 413×5×5

= 13×13×15×13×15×13

= 10813 cu cm

Introduction

Here we find volume of solids made of cubes with unit fraction edge lengths. Consider for example a solid of dimensions 3 in × 3 in × 3 made of small cubes with 12 inch edge lengths.

In that case the solid is made up of 6 × 6 × 6 small cubes of 12 inch edge lengths. So the volume of the solid in this case would be

Volume = l w h = 6×12×6×12×6×12

= 3 × 3 × 3 = 27 cubic inches

Formula for the volume of solid made of cubes with unit fractional edge lengths

Assuming the solid to be a cube of edge a units

b = number of cubes with unit fractional edge length along each edge

k = unit fractional edge length

Volume of solid = b × k × b × k × b × k cubic units

Example 1

Find the volume of following solid of cubes with unit fraction edge lengths. Each prisms unit is measured in cm (not to scale)

Solution

Step 1:

Solid of cubes with unit fraction edge lengths of 12 cm

Step 2:

Volume V = l w h = 212×212×212

= 5×12×5×12×5×12

= 1558 cu cm

Example 2

Find the volume of following solid of cubes with unit fraction edge lengths. Each prisms unit is measured in cm (not to scale)

Solution

Step 1:

Solid of cubes with unit fraction edge lengths of 13 cm

Step 2:

Volume V = l w h = 413×413×413

= 13×13×13×13×13×13

= 811027 cu cm

Introduction

A unit cube has a length of 1 unit, width of 1 unit and height of 1 unit. Its volume is 1 cubic unit.

Any solid shape can be shown as being of made of unit cubes. Counting the number of unit cubes in that solid shape gives its volume. The cube shaded green is a unit cube.

Definition of volume: The number or amount of cubic units it takes to fill a defined space is called volume.

Formula for the volume of a rectangular prism

The volume of a rectangular prism of length l, width w and height h is given by the formula.

The volume of a rectangular prism is given by the product of length, width and the height h of the prism.

Volume V = l × w × h cubic units

Example 1

Find the length, width and height of the following rectangular prism. Then find its volume.

Solution

Step 1:

The given prism has length = 2; width = 3; height = 2

Step 2:

The volume of prism V = l × w × h

= 2 × 3 × 2

= 12 cubic units.

Example 2

Find the length, width and height of the following rectangular prism. Then find its volume.

Solution

Step 1:

The given prism has length = 4; width = 3; height = 2

Step 2:

The volume of prism V = l × w × h

= 4 × 3 × 2

= 24 cubic units.

Definitions

Prism

A prism is a solid bounded by a number of plane faces; its two faces, called the ends or bases, are congruent parallel plane polygons and other faces, called the side faces (or lateral faces), and are rectangles.

Rectangular Prism

In a rectangular prism the bases or ends are rectangular faces and the lateral faces or side faces are also rectangles.

Formula for the volume of a rectangular prism

The volume of a rectangular prism of length l, width w and height h is given by the formula

The product of the length and width of the base of the rectangular prism gives the area of the base A = l × w

The volume of the rectangular prism is given by the product of area of base A and the height h of the prism.

Volume V = A × h = l × w × h cubic units

Example 1

Find the volume of the given rectangular prism in cubic cm.

Solution

Step 1:

Volume of rectangular prism = l × w × h cubic units.

Step 2:

l = 9 cm; w = 3 cm; h = 9 cm

Volume of given prism = 9 × 3 × 9

= 243 cubic cm.

Example 2

Find the volume of the given rectangular prism in cubic cm.

Solution

Step 1:

Volume of rectangular prism = l × w × h cubic units

Step 2:

l = 5 cm; w = 6 cm; h = 4 cm

Volume of given prism = 5 × 6 × 4

= 120 cubic cm.

Definitions

Solid shapes have attributes like face, vertices, edges that describe them.

Individual flat surfaces are called faces. Faces meet in straight lines called edges and points called vertices.

Edges − The line segments that form the skeleton of the 3D shapes are called edges.

Faces − Flat surfaces enclosed by edges are called faces. A face of a solid shape is a plane or 2d figure.

Vertices − The corners where the edges meet are called vertices.

A rectangular prism has 6 faces 8 vertices and 12 edges.

A square pyramid has 5 faces 5 vertices and 8 edges.

Example 1

Determine the faces, vertices and edges of the following shape.

Solution

Step 1:

The given shape is a square pyramid.

Step 2:

It has 5 faces, 5 vertices, and 8 edges.

Example 2

Determine the faces, vertices and edges of the following shape.

Solution

Step 1:

The given shape is a cylinder.

Step 2:

It has 2 faces, 0 vertices, and 0 edges.

Definitions

Solid figures are classified into prisms, pyramids, cylinders, spheres and so on.

A prism is a solid that has two congruent bases which are parallel faces. The other faces of the prism are in the shape of rectangles. They are called lateral faces. A prism is named after the shape of its base.

The following diagrams show a triangular prism and a rectangular prism.

A right prism is a prism that has its bases perpendicular to its lateral surfaces.

A Rectangular prism is a prism that has rectangles as its bases. A shoe box, a soap box, a book, etc., are examples of rectangular prism.

A Cube is a special case of a rectangular prism where all faces and bases are congruent squares.

A dice, ice cubes, Rubiks cube and so on are real world examples of cube shape.

A triangular prism is a prism that has triangles as its bases.

Prisms are named after the shape of their bases. A pentagonal prism has pentagons as its bases, a hexagonal prism has hexagons as its bases and so on.

Cylinder

A Cylinder is a closed solid figure having two parallel circular bases connected by a rectangular curved face.

A Sphere is solid round figure where every point on its surface is equidistant from its centre.

A Cone is a closed solid figure whose circular base tapers into a point called apex.

Pyramids

A pyramid is a solid figure that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex. These triangular sides are called the lateral faces.

Example 1

Use the diagram to identify the name of the following shape.

Solution

Step 1:

The given figure is a prism whose bases are congruent rectangles.

Step 2:

The given figure is a rectangular prism.

Example 2

Use the diagram to identify the name of the following shape.

Solution

Step 1:

The given figure is a solid figure whose bases are congruent circles.

Step 2:

The given figure is a cylinder