3D Figures and Volumes

This tutorial provides comprehensive coverage of 3d Figures and Volumes based on Common Core (CCSS) and State Standards and its prerequisites. Students can navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs. This simple tutorial uses appropriate examples to help you understand 3d Figures and Volumes in a general and quick way.

• Audience

This tutorial has been prepared for beginners to help them understand the basics of 3d Figures and Volumes. After completing this tutorial, you will find yourself at a moderate level of expertise in 3d Figures and Volumes, from where you can advance further.

• Prerequisites

Before proceeding with this tutorial, you need a basic knowledge of elementary math concepts such as number sense, basic arithmetic operations, whole numbers, fractions, decimals, geometric concepts like plane shapes, areas, volumes, nets of solids, unit cubes and so on.

• Classifying Solids
• Vertices, Edges, and Faces of a Solid
• Volume of a Rectangular Prism
• Volume of a Solid made of Cubes with Unit Fraction Edge Lengths

• Word Problem Involving the Volume of a Rectangular Prism

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.

• Word Problem Involving the Rate of Filling or Emptying a Rectangular Prism

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

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

• 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

Volume V = A × h

Where A = 

12

$\frac{1}{2}$ b h or s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−

√

$\sqrt{s(s-a)(s-b)(s-c)}$ or 
23–√

/4

$a^2\sqrt{3}/4$

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

3–√/4

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

a is the side of an equilateral triangle

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