Cone

cone is a solid with a circular base and sides that taper up towards a vertex. A cone is generated from rotating a right triangle, around one leg. A cone has a slant height.

cone.png

Surface area

Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. The basic unit of area is the square unit. For the surface area of a cone we need the sum of the area of the base and the area of the side.

cone s tot

Surface Area of a Right Cone:

conosvil.png

The surface of the base is the surface of a circle and you can use the formula:

cone base

The lateral surface is:

cone s lat

Inversae formulas:

cone inv sl2cone inv sl1

Volume of a cone

Remeber: if you want to find the volume of any solid you must figure out how much space it occupies. The basic unit of volume is the cubic unit.

cone

cone vol

Inversae formula:

cone vol inv

Exercises

  1. The radius of a cone is 5 inches and the volume is 100π cubic inches. Find the slant height and surface area of this cone. Solution
  2. In a cone, the radius r is given as 2 cm and the slant height s as 6 cm. Find its surface area and volume. Solution
  3. A cone has a slant heightof 8 inches and a surface area of 48π square inches. What are the volume and radius of this cone?
  4. A cone has a height of 6 cm and a volume of  V = 8π cm3. What are the radius and      surface area of this cone?
  5. What is the base area of a cone with radius 9 in. and height 46 in.?
  6. What is the base area of a cone with radius 7 in. and height 52 in.?
  7. Hanna distributed paper hats to children on a Christmas eve. The hats are in the shape of a cone with base radius of 8 cm and a slant height of 18 cm. Find the area of the paper used to make a hat.
  8. A circular cone is 9 in. high. The radius of the base is 12 in. What is the lateral surface area of the cone? Solution
  9. The circumference of the base of a conical tent is 43.96 m and its slant height is11 m. Find the area of the canvas used in making the tent.
  10. A paper cone is 48 cm high and has a radius of 20 cm. Find the area of the paper needed to make the cone.
  11. What is the base area of a cone with radius 6 cm and height 33 cm?
  12. The total surface area of a cone is 240 square cm. If its slant height is four times the radius, then what is the base diameter of the cone?
  13. A circular cone with a base radius of 13 cm has a surface area of 2122.64 cm2.What would be its slant height?

Cylinder

A cylinder is a three-dimensional shape with a pair of parallel and congruent circular ends, called bases.

Cylinders can be right or oblique. The side of a right cylinder is perpendicular to its circular bases.

cylinder

A cylinder has a single curved side that forms a rectangle when laid out flat.

cylindersviluppo

Surface area of a cylinder

cylsurf.png

The area of each base is given by the area of a circle:

areacerchio.png

The area of the rectangular lateral area is given by the product of a width and height. The height is given as h; you can see that the width of the area is equal to the circumference of the circular base.

The circumference of a circle is given by:

circonferenza

Lateral surface is:

suplatcilindro

Inversae formulas:

suplatcilindroinv1       suplatcilindroinv2

Total surface area is:

suptotprisma

Volume of a right cylinder

Right prisms and right cylinders are very similar with respect to volume. In a sense, a cylinder is just a “prism with round bases.”

The volume of a right cylinder with radius r and height h can be expressed as:

volume cil

Inversae formulas:

volume cilinv1         volume cilinv2

Exercises:

  1. Find the volume of the following cylinder:

cil1

2. The radius of the base of a cylinder measures 12 cm and its height exceeds 5 cm three times the radius. Calculate the total surface area and the vol ume of the solid.

3. The lateral surface of a cylinder is of 471 square centimeters. Calculate the volume of the solid knowing that is 15 cm high.

4. The height of a cylinder is 6 cm and the volume is 121.5 pi cubic centimeters. Calculates the lateral surface of the solid.

5. A wedding cake is made of three overlapping cylinders whose diameters are  80 cm, 40 cm and 2 cm and the height of the three cylinders is 12 cm. Find    the total surface area and volume of the cake.

6. In a cylinder is dug prism a square based. Knowing that the height of the cylinder and of the prism measures 20 cm and the radius of the cylinder is 12 cm and the side of a square measures 6 cm, find the total surface area and the volume of the solid.

7. You want to dig a tunnel 12 meters long and 3.5 meters high having the shape of a half cylinder. Calculate the volume of material which must be removed.

8. Calculates the measurement of the radius of a cylinder, knowing that the lateral area is 301.44 squared centimeters and height measures 15 cm.

Pyramid

 

A pyramid is a three dimensional shape made up of a base and triangular faces that meet up at the vertex, V, which is also called the apex of the pyramid.

pyramid

Surface area of a pyramid

The surface area is composed by two parts: the area of the base and the lateral area.

Area of the base

It depends on the shape, you have to use the correct formula if the base is composed by a triangle, square, ecc.

Lateral surface

Since the side faces are triangles and the formula for the area of a triangle is

area triangolo

All you need to do is multiply the area of the base by the slant lenght and divide by two

area laterale piramide

Inverse formulae:

area laterale piramide inv2

area laterale piramide inv1

Volume of the pyramid

volume piramide.png

Inverse formulae:

volume piramide inv2

volume piramide inv1

Exercises

  1. A pyramid has slant lenght 10 cm long. The base is a square with a side of 7 cm. Calculate the total surface area. Solution
  2. A pyramid has a slant lenght 125 cm long. The lateral surface is 450 dm2. Calculate the perimeter of the base. Solution
  3. A pyramid has a volume of 1840 cm3. The base is a square that has an area of 1.84 cm2. Calculate the height of the pyramid. Solution
  4. The base of a pyramid is a rhombus, that has an area of 37 cm2. Calculate the lateral surface area knowing that the total surface area is 140.5 cm2.
  5. The base of a pyramid is an isosceles triangle ABC with AB = 10 cm and the height CH = 12 cm. The height of the pyramid is 15 cm. Calculate the volume of the pyramid.

Prism

 

A prism is a solid figure that has two parallel congruent sides that are called bases that are connected by the lateral faces that are parallelograms. There are both rectangular and triangular prisms.

prisma 1prisma2

Surface area of a Prism

If we want to find the total surface of a prism we need to add twice the surface of the base plus the lateral surface.

suptotprisma

In order to find the surface of the base you only have to calculate the surface of a bidimensional shape.

The lateral surface is given by the perimeter of the base multiplied by the height:

suplatprisma

Inverse formulas

prisma inverse superficie

Volume

To find the volume of a prism (it doesn’t matter if it is rectangular or triangular or else) we multiply the area of the base, called the base area S, by the height h.

volumeprisma

Inverse formulas:

volumeprismainversa

Exercises

  1. A letter ‘T’ shape is made by sticking together 2 cuboids as shown in the diagram.

What is the total volume in cm³ and the surface area in cm2 of the letter ‘T’?

11.png

2. Calculate the volume and the surface area of each of the cuboids below.

12                     13

14              15

3. A prism has a height of 32 cm. The base is a rhombus and the diagonals are one 3/4 of the other, their difference is 18 cm. Calculate the total surface area of the prism. Solution

4. A prism has a height of 978 cm. The base is a right angled trianglewith the hypotenuse 75 cm long; the hypotenuse is 25/7 of one of the legs. Calculate the total surface area. Solution

5. A prism has a right angled triangle as base. One leg is 27 cm long and the area of the triangle is 486 cm2 . The total surface area of the prism is 7776 cm2 . Calculate the volume and the height of the prism. Solution

6. Calculate the surface area and the volume of the following solids:

prisma es

7. The followimg image shows a prism whose base is a trapezium. Calculate the total surface area of the prism.

prism es 22

8. The following image shows a prism whose base is a trapezium. Given that the total surface area of the prism is 126 cm2 , find the value of x.

prism es 23

 

Cuboid

Cuboid

A rectangular solid is also called a rectangular prism or cuboid.

In a rectangular solid, the lenght, width and height might be of different lenghts

Volume

The volume is found using the formula:

Inverse formulas:

Surface area

The total surface area (TSA) of a cuboid is the sum of the areas of its six faces.
That is:

For  example:

Find the total surface area of a cuboid with dimensions 8 cm by 6 cm by 5 cm.
TSA = 2 (lw+wh+hl)
TSA = 2 (8⋅6+6⋅5+5⋅8) = 2(48+30+40) = 236 cm2
So, the total surface area of this cuboid is 236 cm2
 
 

For  example:

We want to find the volume of this cuboid in cm3

To find the volume we mulptiply the sides.

V=3∙4∙7 = 84 cm3

Exercises

  1. A cuboid has lateral surface 199 cm . The base has sides 23 cm and 14 cm long. Calculate the height. Solution
  2. A cuboid has its height equal to the diagonal of the base. The sum of the two sides of the base is 21 cm and one is 3/4 of the other. Calculate the total surface area.
  3. The diagonal of a cuboid is 117 cm the base has sides 36 cm and 27 cm long. Calculate the volume and the total surface area.
  4. A cuboid has the perimeter of the base equal to 84 dm and one side is 3/4 of the other. The height is 40 dm long. Calculate the total surface area.
  5. The sum of two sides of a cuboid is 14 cm and one side is 4/3 of the other. Knowing that the total surface area is 768 cm, calculate the third dimension of the cuboid.
  6. The difference between two sides of a cuboid is 14 cm and one side 15/8 of the other. The third side (height) is 33 cm long. A second cuboid has a squared base with 121 cmarea and has the same lateral area of the first one. Calculate the third side.
  7. A cuboid has two sides 13 cm and 15 cm long. The total surface area is 950 cm. Calculate the volume. Solution
  8. The squared base of a cuboid is 576 cm. The height of the cuboid is 1/12 of the perimeter of the base. Calculate lateral surface and volume.
  9. The volume of a cuboid is 108 cm3 and two sides are 4.5 cm and 6 cm long. Calculate the diagonal and the total surface area of the cuboid.
  10. The volume of a cuboid is 1768 cm3 . The sum and the difference between two sides are 30 cm and 4 cm. Calculate the lenght of the third side.
  11. The perimeter of the base of a cuboid is 66 cm and one side is 6/5 of the other. The volume is 2.97dm3 . Calculate the lenght of the third side.

Cube

Solid Geometry

Three-dimensional shapes

Cube

A cube is a three-dimensional figure with six matching square sides.
The dotted lines indicate edges hidden from your view.

Volume

If s is the length of one of its sides, then the volume of the cube is s · s · s
Volume of the cube

Inverse formula:

Surface area

The area of each side of a cube is s2.
Since a cube has six square-shape sides, its total surface area is 6 times s2.
Surface area of a cube

Inverse formula:
<!– [if gte msEquation 12]>s=TSA6<![endif]–>

For example:
The length of the cube’s side is 3.8 cm.
Volume = S3 (Substitute the length in formula)
= 3.83 (Multiply the length three times)
= 3.8 · 3.8 · 3.8 = 54.87 cm3
For example:
A cube shape box is filled with small balls and the length of box is 21 cm. What is the volume of cube box?
Volume = S3 (Substitute the length of box in formula)
= 213 (Expand the value in three)
= 21 · 21 · 21 (Multiply the value 21 three times)
= 9261 cm3
For example:
The side of a cube is 5cm. Find its total surface area.
Total surface area = 6a2, where a is side.
Given that a = 5cm.
Total surface area of the cube = 6 · 52
= 6 · 25
= 150 cm2.
For example:
Find the surface area of a cube with a side of length 3 cm.
Given that s = 3
Surface area of a cube = 6s2
= 6(3)2
= 54 cm2

Exercises

1. Find the volume and the total surface area of a cube with sides 8 cm.
2. If the side of a cube is 3.5 inches, find its volume and its total surface area.
3. What is the volume and the total surface area of a cubical box with sides 7 inches?
4. Find the volume and the total surface area of a cube with sides 1.7 feet.
5. Dimensions of a cube: Length = Width = Height = 5.6 inches. Find the volume and the total surface area of the cube.
6. A cube has the side 18 cm long. Calculate the volume and total surface area of the cube.
7. A cube has the side 25cm long . Calculate the volume and the total surface area.
8. The total surface area of a cube is 9126 cm ². Calculate the side.
9. The volume of a cube is 3375 cm3. Calculate the total surface area of the cube.
10. The perimeter of one side of a cube is 120 cm. Calculate the volume and total surface area of the cube. Solution
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