Where r is the radius.
Example Calculate the volume of a sphere with radius 6cm.
This can be done on a calculator by scientific calculator, use 3.14)
, using the
or if you don’t have a
Volume of a cone The formula for calculating the volume of a cone, where r is the radius and h is the perpendicular height is:
Example Calculate the volume of a cone with radius 5cm and height 12cm.
Answer
Volume of a prism A prism is a solid with a uniform cross section. This means that no matter where it is sliced along its length, the cross section is the same size and shape (congruent).
A well-known example of a prism is a cylinder and you can see from the image above that the front face (cross section) is the same size of circle no matter where you slice it. The formula for the volume of a prism where height/length of the solid is:
is the area of the cross section and is the
Example This shape is a triangular prism so the area of the cross section is the area of a triangle.
Answer Area of the triangle:
Volume of the prism:
Volume of a cylinder
The formula for the volume of a cylinder (circular prism) is derived from the volume of a prism, where is the radius and is the height/length.
Since the area of a circle =
, then the formula for the volume of a cylinder is:
Example Calculate the volume of the cylinder shown.
Give your answer correct to 1 significant figure.
Answer
Volume of a hemisphere
Half a sphere is called a hemisphere.
Example A glass bowl is in the shape of a hemisphere with diameter 13cm.Trisha will fill the bowl with water so that she can use it for floating candles. What is the maximum amount of water the glass bowl can hold? Give your answer in millilitres correct to 2 significant figures.
Answer Diameter = 13cm therefore the radius =
.
Volume of Sphere =
Since
, the glass bowl can hold 580 ml (to 2 s.f.)
Volume of composite shapes To calculate the volume of a composite shape, simply split it into smaller shapes and calculate their separate volumes. The volumes of each of the individual shapes are then added together to give the total volume of the composite shape.
Example Calculate the volume of the shape shown. Give your answer correct to 2 significant figures.
Answer Diameter = 10m therefore the radius = Volume of cylinder:
Volume of sphere:
Volume of hemisphere
A formula triangle involving force, mass, and acceleration. Cover one up and it gives you the formula that you need to find the one you're hiding!
Acceleration is the rate at which an object changes its speed. It's calulated using the equation: acceleration = change in speed / time taken. Speed-time graphs illustrate how the speed of an object changes over time. The steeper the gradient of the line, the greater the acceleration.
Acceleration In everyday language we use 'accelerate' to mean speeding up and 'decelerate' to mean slowing down. In scientific terms 'acceleration' is the rate at which something changes its speed - faster or slower. Acceleration depends on two things:
How much the speed changes
How much time the change in speed takes Calculating acceleration Acceleration = change in speed / time taken
Example A bus accelerates from 5 m/s to 25 m/s in 10s To calculate its acceleration, first find the change in speed.
This is 25m/s - 5m/s = 20m/s Acceleration = 20m/s ÷ 10s = 2m/s2
Use this triangle to help you rearrange the equation to:
change in speed = acceleration x time taken time taken = change in speed / acceleration
Work and power Page: 1.
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2.
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Whenever 'work' is done energy is transferred from one place to another. The amount of work done is expressed in the equation: work done = force x distance. Power is a measure of how quickly work is being done. Power is expressed in the equation: power = work done / time taken.
Work and force Work done Work is done whenever a force moves something.
Everyday examples of work include walking up stairs, or lifting heavy objects. Whenever work is done energy is transferred from one place to another. Both energy and work are measured in joules, J. work done (joules, J) = energy transferred (joules, J) The amount of work done depends on:
the size of the force on the object
the distance the object moves
Equation work done (joule, J) = force (newton, N) x distance (metre, m) How much work is done when a man lifts a box weighing 200N off the floor to a shelf 2m high? Work done = force x distance = 200N x 2m = 400J
Higher tier
Use the triangle to help you rearrange the equation to: force = work done / distance distance = work done / force
Rearranging equation activity - higher tier only Check your understanding of this by having a go at the activity.
Power – work and time Power is a measure of how quickly work is being done and so how quickly energy is being transferred. More powerful engines in cars can do work quicker than less powerful ones. As a result they usually travel faster and cover the same distance in less time but also require more fuel.
Car comparison
Car A (standard)
Car B (sports)
Power
44
240
Top speed (km/h)
160
285
Fuel consumption (litres/100km)
6
11
Question If both fuel tanks hold 50 litres how far could each car drive without refuelling?
Answer Car A 100km x 50/6 = 833km Car B 100km x 50/11 = 455km (to nearest km)
Higher tier power (watts, W) = work done (joule, J) / time taken (seconds, s)
Question What is the power of an engine that does 3000J of work in 60s?
Answer Power = work done / time taken Power = 3000J/60s = 50W
Higher tier Use the triangle to help you rearrange the equation to:
The equation All of the calculations in this section will be worked out using the distance, speed and time equation.
An easy way to remember the distance, speed and time equations is to put the letters into a triangle.
The triangles will help you remember these 3 rules:
On the next page there are some examples to work through. Have paper and a pen handy, draw the distance, speed and time triangle on your paper, then try the examples
Example Iain walked from his parents' farm into town at a steady speed of The journey took
.
. How far did Iain walk?
In the first hour he walked
.
After two hours he had walked After three hours he had walked
. .
Now you can try this example Shona cycles at an average speed of ?
. How far has she travelled if she cycles for
Shona has travelled
.
Units It is important that, for all of these calculations, the units used correspond with each other. If the distance is given in kilometres and the time in hours, then the measurement of speed should be given in the form of kilometres per hour. This is written as km/h. This next question shows where you need to be careful with units Kelly runs from
until
However, the speed is given in
at an average speed of
. How far did she go
, so our time must be given in hours.
Kelly ran
.
Calculating speed, given distance and time Alan travels
in
. Find his average speed in
. Alan's average speed is Find the speed of a train which travels
. in
.
The train is travelling at Joanna drives for
at an average speed of
Joanna's journey was
long
. How long was her journey
