Solutions
Find the area of each of the polygons below:
The radius is half the diameter, therefore the radius is .
We first need to work out the height using the theorem of Pythagoras:
Now we can calculate the area:
We first need to work out the height using the theorem of Pythagoras:
Now we can calculate the area:
We first need to construct the vertical (or perpendicular) height. For an isosceles triangle if we construct the perpendicular height between the two equal sides then this line will bisect the third side.
Now we can calculate the height using the theorem of Pythagoras:
Now we can calculate the area:
We first find the height using the theorem of Pythagoras:
Now we can calculate the area:
Calculate the surface area of the following prisms:
There are three different sized rectangles that make up the sides of this triangular prism. We need to find the area of each one of them. All of the rectangles have a height of 11 but each rectangle has a different base.
There are 4 rectangles and 2 squares that make up this rectangular prism. The square has a side length of 5. The rectangles have a base of 5 and a height of 11.
We first need to find the missing side of the triangle. We can do this using the theorem of Pythagoras.
Now we can find the area of the triangular prism:
If a litre of paint covers an area of , how much paint does a painter need to cover:
a rectangular swimming pool with dimensions (the inside walls and floor only);
We need to find the surface area of the pool. In this case we have a rectangular prism but with one rectangle missing (which would be the top of the pool).
The painter needs one litre of paint for every of area. So we must divide the surface area by 2 to find the total amount of paint needed. Therefore, the painter will need of paint (rounded up to the nearest litre).
We need to find the surface area of the reservoir. In this case we have a cylinder but with one circle missing (which would be the top of the reservoir).
We are given the diameter of the reservoir. The radius is half the diameter and so .
The painter needs one litre of paint for every of area. So we must divide the surface area by 2 to find the total amount of paint needed. Therefore, the painter will need of paint (rounded up to the nearest litre).
Calculate the volumes of the following prisms (correct to decimal place):
The figure here is a triangular prism. The height of the prism is units; the triangles, which both contain right angles, have sides which are , and units long. Calculate the volume of the figure. Round to two decimal places if necessary.
The figure here is a rectangular prism. The height of the prism is units; the other dimensions of the prism are and units. Find the volume of the figure.
The picture below shows a cylinder. The height of the cylinder is units; the radius of the cylinder is units. Determine the volume of the figure. Round your answer to two decimal places.
Find the total surface area of the following objects (correct to 1 decimal place if necessary):
We first need to find by constructing the vertical (perpendicular) height and using the theorem of Pythagoras:
Now we can find the surface area:
The figure here is a cone. The vertical height of the cone is units and the slant height of the cone is units; the radius of the cone is shown, units. Calculate the surface area of the figure. Round your answer to two decimal places.
Therefore the surface area for the cone is square units.
The figure here is a sphere. The radius of the sphere is units. Calculate the surface area of the figure. Round your answer to two decimal places.
Therefore the surface area is square units.
The figure here shows a pyramid with a square base. The sides of the base are each units long. The vertical height of the pyramid is units, and the slant height of the pyramid is units. Determine the surface area of the pyramid.
The surface area for the pyramid is square units.
The figure below shows a sphere. The radius of the sphere is units. Determine the volume of the figure. Round your answer to two decimal places.
Therefore the volume for the sphere is units^{3}.
The figure here is a cone. The vertical height of the cone is units and the slant height is units; the radius of the cone is shown, units. Calculate the volume of the figure. Round your answer to two decimal places.
Therefore the volume of the cone is units^{3}.
The figure here is a pyramid with a square base. The vertical height of the pyramid is units and the slant height is units; each side of the base of the pyramid is units. Round your answer to two decimal places.
Therefore the volume of the square pyramid is: units^{3}.
Find the volume of the following objects (round off to 1 decimal place if needed):
We are given the radius of the cone and the slant height. We can use this to find the vertical height () of the cone:
Now we can calculate the volume of the cone:
We first need to find the vertical (perpendicular) height of the triangle () using the theorem of Pythagoras:
Now we can find the volume:
The surface area of the cone is:
For the volume we first need to find the perpendicular (or vertical) height using the theorem of Pythagoras:
Now we can calculate the volume of the cone:
Therefore the surface area is and the volume is .
Calculate the following properties for the pyramid shown below. Round your answers to two decimal places.
Surface area
We first calculate the vertical (perpendicular) height of the base triangle:
Now we can calculate the surface area of the pyramid:
Therefore the surface area of the triangular pyramid is: .
Volume
We first need to find the vertical height ():
Now we can find the volume:
Therefore the volume of the pyramid is: .
The solid below is made up of a cube and a square pyramid. Find its volume and surface area (correct to decimal place):
The height of the cube is . Since the total height of the object is , the height of the pyramid must be .
We will work out the volume first:
For the surface area we note that one face of the cube is covered by the pyramid. We also note that the base of the pyramid is covered by the cube. So we only need to find the area of 5 sides of the cube and the four triangular faces of the pyramid.
For the triangular faces we need the slant height. We can calculate this using the theorem of Pythagoras:
The surface area is:
The surface area is and the volume is .
If the length of the radius of a circle is a third of its original size, what will the area of the new circle be?
The area of the original circle is: . Now we reduce the radius by a third. In other words we multiply by one third. The new area is:
Therefore, if the radius of a circle is a third of its original size, the area of the new circle will be the original area.
If the length of the base's radius and height of a cone is doubled, what will the surface area of the new cone be?
We can find the new area by noting that the area will change by a factor of when we change the dimensions of the cone. In this case we are changing two dimensions of the cone and so the new area will be:
The value of comes from the word “doubled” in the question: the value of is 2.
So the new area of the cone will be if we double the height and the base's radius of the cone.
Therefore the surface area of the new cone will be 4 times the original surface area.
If the height of a prism is doubled, how much will its volume increase by?
We do not know if we have a rectangular prism or a triangular prism. However we do know that the volume of a prism is given by:
Now we are changing just one dimension of the prism: the height. Therefore the new volume is given by:
Therefore the volume of the prism doubles if the height is doubled.
Describe the change in the volume of a rectangular prism if the:
length and breadth increase by a constant factor of .
The volume of a rectangular prism is given by . If we increase the length and breadth by a constant factor of 3 the volume is:
Therefore the volume of the prism increases by a factor of 9 when the length and breadth are increased by a constant factor of 3.
length, breadth and height are multiplied by a constant factor of .
The volume of a rectangular prism is given by . If we increase the length, breadth and height by a constant factor of 3 the volume is:
Therefore the volume of the prism increases by a factor of 27 when the length, breadth and height are increased by a constant factor of 3.
If the length of each side of a triangular prism is quadrupled, what will the volume of the new triangular prism be?
When multiplied by a factor of the volume of a shape will increase by . We are told that the dimensions are quadrupled. This means that each dimension is multiplied by 4. Therefore .
Now we can calculate .
Therefore, if each side of a triangular prism is quadrupled, the volume of the new triangular prism will be 64 times the original shape's volume.
Given a prism with a volume of and a surface area of , find the new surface area and volume for a prism if all dimensions are increased by a constant factor of .
We are increasing all the dimensions by 4 and so the volume will increase by . The surface area will increase by .
Therefore the volume is and the surface area is .
Find the area of each of the shapes shown. Round your answer to two decimal places if necessary.
We first need to find the height:
Now we can find the area of the parallelogram. Note that the length of the base is .
The figure here is a triangular prism. The height of the prism is units; the triangles, which are both right triangles, have sides which are , and units long. Find the surface area of the figure.
A triangular prism is made up of 2 triangles and 3 rectangles. In this case the triangles are rightangled triangles and so we have the height of the triangle. We also note that each rectangle has a different length and breadth.
The figure here is a rectangular prism. The height of the prism is units; the other dimensions of the prism are and units. Find the surface area of the figure.
A rectangular prism is made up of 6 rectangles. In this case there are 4 rectangles with a breadth of 5 units and a height of 8 units and two rectangles with a breadth of 5 units and a height of 5 units.
A cylinder is shown below. The height of the cylinder is ; the radius of the cylinder is , as shown. Find the surface area of the figure. Round your answer to two decimal places.
A cylinder is made up of two circles and a rectangle. We can find the area of each of these and add them up to find the surface area of the cylinder. For the rectangle we note that the length is the circumference of the circle.
The figure here is a triangular prism. The height of the prism is units; the triangles, which both contain right angles, have sides which are , and units long. Determine the volume of the figure.
The figure here is a rectangular prism. The height of the prism is units; the other dimensions of the prism are and units. Calculate the volume of the figure.
The picture below shows a cylinder. The height of the cylinder is ; the radius of the cylinder is . Calculate the volume of the figure. Round your answer two decimal places.
The figure here is a sphere. The radius of the sphere is units. Find the surface area of the figure. Round your answer two decimal places.
The figure here shows a pyramid with a square base. The sides of the base are each units long. The vertical height of the pyramid is units, and the slant height of the pyramid is units. Determine the surface area of the pyramid.
The total surface area for the pyramid is: square units.
The figure here is a cone. The vertical height of the cone is units and the slant height of the cone is units; the radius of the cone is shown, units. Find the surface area of the figure. Round your answer two decimal places.
Therefore the total surface area for the cone is square units.
The figure below shows a sphere. The radius of the sphere is units. Determine the volume of the figure. Round your answer to two decimal places.
Therefore the volume for the sphere is units^{3}.
The figure here is a cone. The vertical height of the cone is units and the slant height is units; the radius of the cone is shown, units. Find the volume of the figure. Round your answer to two decimal places.
Therefore the volume is units^{3}.
The figure here is a pyramid with a square base. The vertical height of the pyramid is units and the slant height is units; each side of the base of the pyramid is units. Find the volume of the figure. Round your answer to two decimal places.
Therefore the volume is: units cubed
Calculate the surface area of each solid.
Cone
We first need to calculate the slant height:
Now we can calculate the surface area:
Square pyramid
We first need to calculate the slant height:
Now we can calculate the surface area:
Half sphere
For a half sphere we need to divide the surface area of a sphere by 2. We also need to include the area of a circle.
The surface area of each of the objects is: .
Calculate the volume of each solid.
Cone
Square pyramid:
Half sphere
The volume of a half sphere is half the volume of a sphere.
The volume of each of the objects is:
If the length of each side of a square is a quarter of its original size, what will the area of the new square be?
When we multiply the sides of a square by a factor of the area of the square will increase by .
In this case we are making each side of the square a quarter of the original size so we get:
Therefore, if each side of a square is a quarter of its original size, the area of the new square will be times the original square's area.
If the length of each side of a square pyramid is a third of its original size, what will the surface area of the new square pyramid be?
When we multiply two dimensions of a square pyramid by a factor of the area of the square pyramid will change by .
In this case the length each side of the square pyramid is a third of the original size so we get:
Therefore, if each side of a square pyramid is a third of its original size, the surface area of the new square pyramid will be times the original shape's surface area.
If the length of the base's radius and the height of a cylinder is halved, what will the volume of the new cylinder be?
In this case the base's radius and the height of a cylinder is half of the original size so we get:
Therefore, if the base's radius and the height of a cylinder is halved, the volume of the new cylinder will be times the original shape's volume.
Consider the solids below and answer the questions that follow (correct to decimal place, if necessary):
Calculate the surface area of each solid.
Cylinder
A cylinder is composed of two circles and a rectangle. The breadth of the rectangle is the circumference of the circle.
Triangular prism
A triangular prism is composed of three rectangles and two triangles. We are given the vertical height of the triangles as well as the slant height.
Rectangular prism
A rectangular prism is composed of 6 rectangles. We have the dimensions of all the rectangles.
The surface area of each shape is: .
Calculate volume of each solid.
Cylinder
Triangular prism
Rectangular prism
The volume of each shape is: .
If each dimension of the solids is increased by a factor of , calculate the new surface area of each solid.
Cylinder
Triangular prism
Rectangular prism
The new surface area of each shape is: .
If each dimension of the solids is increased by a factor of , calculate the new volume of each solid.
Cylinder
Triangular prism
Rectangular prism
The new volume of each shape is:
Find the surface area of the solid shown. Give your answers to two decimal places.
Start with the faces of the cube, which are all squares:
Next we note that the height of the pyramid is:
And we need to calculate the slant height using the theorem of Pythagoras:
Now we can calculate the area of each of the four triangles:
Finally we can calculate the total surface area:
Therefore the surface area is: .
Now determine the volume of the shape. Give your answer to the nearest integer value.
Volume of the pyramid:
Volume of the cube:
Total volume:
Therefore the total volume is: .
Surface area
Cylinder:
Cone:
Total surface area: .
Volume
Cylinder:
Cone:
Total volume .
The total surface area and volume is and respectively.
Find the volume and surface areas of the following composite shapes.
The shape is a half sphere on top of a right cone. We can calculate the volume of a cone and add this to half the volume of a sphere. The volume is:
For the surface area we first need to find the slant height:
We have a half sphere on top of a cone. The half sphere covers the circle on top of the cone and so we need to exclude this part from our calculation. For the half sphere we can use half the surface area of a sphere as this does not include the circle at the base of the half sphere.
The surface area is:
Therefore the volume and surface area are and respectively.
We have a cylinder with two half spheres. We can calculate the volume of a cylinder and add the volume of a sphere to this. The volume is:
For the surface area of the two half spheres we can use the surface area of a sphere. For the cylinder we need to exclude the area of the two circles from our calculation since these are covered up by the two half spheres. The surface area is:
Therefore the volume and surface area are: and respectively.
This shape consists of a triangular prism and a rectangular prism. The volume is:
For the surface area we need to exclude the base of the triangular prism as well as part of the top of the rectangular prism.
We first need to calculate the slant height for the triangular prism:
Now we can calculate the surface area of the triangular prism. Remember that we do not need to include the base in our calculation so we only have 2 triangles and 2 rectangles.
For the rectangular prism we can calculate the full surface area and then subtract the base of triangular prism from this.
Now we can add the two surface areas together to get the total surface area:
The volume and surface area are and respectively.
An icecream cone (right cone) has a radius of and a height of . A half scoop of icecream (hemisphere) is placed on top of the cone. If the icecream melts, will it fit into the cone? Show all your working.
We can draw a quick sketch of the problem:
Now we can calculate the volume of the cone and the volume of the icecream. The scoop of icecream is a half sphere and so the volume of this is half the volume of a sphere.
Yes, the icecream will fit into the cone if it melts since the volume of the icecream is less than the volume of the cone.
A receptacle filled with petrol has the shape of an inverted right circular cone of height and base radius of . A certain amount of fuel is siphoned out of the receptacle leaving a depth of cm.
Show that .
Determine the volume of fuel that has been siphoned out. Express your answer in litres if
The volume of fuel that has been siphoned out is the total volume of fuel minus the volume of fuel left. The volume of a cone is . From the previous question we have the vertical height for both cones.
Find the volume and surface area of the following prisms.
We are given the diameter of the cylinder. The radius is half the diameter.
Therefore the volume and surface area are and respectively.
This is a triangular pyramid. We are given the vertical height as well as an angle. Since it is a rightangled triangle we can use trigonometry to help us find the missing length.
We redraw the triangle we are interested in:
Now we can calculate (the slant height) and (the base):
Now we know all the lengths we need to know to calculate the volume.
And the surface area is:
The volume and surface area are: and respectively.
Let: , , , , and .
We can view this shape as three rectangular prisms. Two of the three prisms are exactly the same. The volume is therefore:
For the surface area we have several different rectangles. Each of the smaller prisms has 5 exposed rectangles. The larger rectangular prism has 4 rectangles that are not covered up by the smaller prisms. The remaining two rectangles are partly covered up by the smaller prisms and so can be considered as 4 separate rectangles.
We will start by finding the surface area of one of the smaller prisms:
For the larger prism we get:
Therefore the total surface area is:
The volume and surface area are and 886 respectively.
Determine the volume of the following:
We first need to find the vertical height ():
The prism alongside has the following dimensions:
Explain why , the radius of the arc , is units.
Since is the centre of the circle (they are both radii of the arc).
and joins and , therefore .
We also know that and since , . Therefore is 4 units.
Calculate the area of the shaded surface.
We have just calculated that . We also know that and so is a square (). This means that we have the area of a square plus one quarter the area of a circle.
The total area is:
Find the volume of the prism.
The area of the shaded piece is the area of the base. For the volume we know that we can calculate the volume by multiplying the area of the base and the height.
You can also calculate the volume using the volume of a rectangular prism and one quarter of the volume of a cylinder.
A cooldrink container is made in the shape of a pyramid with an isosceles triangular base. This is known as a tetrahedron. The angle of elevation of the top of the container is . ; .

Show that the length is cm.

Find the height (to the nearest unit).

Calculate the area of .
Hint: construct a perpendicular line from U to

Find the volume of the container

is a rightangled triangle. We can use trigonometry to help us find . In this case we will use the cosine ratio as we have the hypotenuse () and are looking for the adjacent side ().

is a rightangled triangle. We can use trigonometry to help us find . In this case we will use the sine ratio as we have the hypotenuse () and are looking for the opposite side ().

We first find :
Now we can find the area:
The container is filled with the juice such that an % gap of air is left. Determine the volume of the juice.
To find the volume of the juice we need to multiply the total volume of the container by the percentage of juice in the container.
Below is a diagram of The Great Pyramid.
This is a squarebased pyramid and is the centre of the square.
and . The length of the side of the pyramid and the height of the pyramid is .
Determine the area of the base of the pyramid in terms of .
Calculate to decimal places.
From your calculation in question (b) determine .
Determine the volume and surface area of the pyramid.
The volume and surface area are: and respectively.
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