Shapes - 2D and 3D

The Perimeter of a two-dimensional shape is the length of its outline.

Most of the time you just need to find the length of each side of the shape and add them all up.

Some shapes have formulas for their perimeters: Perimeter of a square with sides a = 4a Perimeter of a rectangle with sides a, b = 2(a + b) Perimeter of a circle with radius r = 2πr

For more complex shapes, be sure to include the length of every side when calculating the perimeter. It might help to mark the sides as you go around the shape.

Calculate the perimeter of the shape below.

First, work out the lengths of the sides that aren't marked. The long vertical = 2.5m + 2.5m = 5m The short horizontal = 4.5m - 2m = 2.5m So, starting at the top left hand corner and going round clockwise Perimeter = 2 + 2.5 + 2.5 + 2.5 + 4.5 + 5 = 19m

It's important to know the names of two-dimensional shapes. Some have various different types eg. triangles and quadrilaterals.

Learn the names, appearance and properties of all the shapes in the table

Also learn what the markings mean (eg. arrows for parallel lines, dashes across lines to show they're equal in length, right angle marks)

How many sides does an octagon have?

8

What's the special name for a quadrilateral with four equal sides but no right angles?

A rhombus (it looks like a squashed square)

A Polygon is a two-dimensional shape with 3 or more sides.

See the notes on Identifying 2D Shapes to learn some of the names of polygons.

In addition, a Nonagon has 9 sides and a Decagon has 10 sides.

To find out how to calculate the interior angles of a regular polygon, see the module Angles and Regular Polygons

Which regular polygon has 4 pairs of parallel sides?

To have 4 pairs of parallel sides, there must be 2 x 4 sides in total so it's a regular octagon.

There are a number of Special Quadrilaterals. Here they are with some of their properties.

Learn the shapes and learn their properties.

Which special quadrilateral has four equal angles and equal diagonals that bisect at right angles?

Squares and rectangles have four equal angles (all 90°). They also have equal diagonals. However, only the square's diagonals bisect at right angles.

The Area of a Rectangle with sides lengths a, b is ab

If in doubt, draw a diagram

The surface area of a trestle table is 3m² and the length of its shorter side is 75cm. What's the length of its longer side?

Let the longer side be length a. So, we know the Area of a Rectangle with sides lengths a, b is ab In this case 0.75a = 3 So a = 4 The length of the longer side is 4m

The Area of a Triangle is given by the formula 1/2 Base x Perpendicular Height

The only potentially tricky bit is making sure the perpendicular height is the one for the base.

Find the area of the triangle below.

At first sight it seems as if there's not enough information. However... The base and the side marked 8cm are the same length (there are dashes on them). So, the area of the triangle = 1/2 x 8 x 7.5 = 30cm²

The Area of a Parallelogram is Base x Perpendicular Height

Just use the formula!

What is the perpendicular height of a parallelogram with base 8cm and area 56cm²

Area = Base x Height So, 56 = 8 x Height Height = 7cm

The Area of a Trapezium = h(a+b)/2 where h is the height and a, b the lengths of the parallel sides.

Use the formula, and if you need to, draw a diagram.

A trapezium's base is 7cm, its height is 5cm and its area is 25cm². What's the length of the shorter side?

Area = h(a+b)/2 Putting in the values we have: 25 = 5(a+7)/2 Expanding and rearranging we get 50 - 35 = 5a This gives 5a = 15 So, the shorter side is 3cm

Compound Shapes are made up of a number of smaller, simpler shapes.

The key is to break the shape down into its constituent parts, work out the area of each part and then sum the areas up.

It's a good idea to draw a diagram showing the constituent smaller shapes.

What is the area of this simple drawing of a ship?

Break the ship down into its constituent parts. Here, the top is a rectangle and the bottom is a trapezium.

The area of the trapezium = (8+16)x7/2 = 84 cm² The area of the rectangle = 8 x 3 = 24 cm² So, the total area = 24 + 84 = 108 cm²

A Cuboid is a rectangular block

Volume is Length x Width x Height Volume = lhw

Surface Area is 2 ( Length x Width + Width x Height + Height x Length) Surface Area = 2 (lw + wh + hl)

A Cube is a special case of a cuboid where Length = Width = Height

Use the formulas and remember to give your answer in the correct units.

A cuboid has two sides that are 3cm and 5cm and its volume is 120cm³ Calculate its surface area

Volume = Length x Width x Height So, 120 = 5 x 3 x h That makes h = 8cm Surface Area = 2 ( lh + hw + wl ) ie. 2 ( 5 x 8 + 8 x 3 + 3 x 5 ) = 158 cm²

A Prism is a solid shape which has a cross-section with a constant area. This means that wherever you slice a prism perpendicular to its length, you'll get the same shape.

A Cylinder is a Circular Prism A Cuboid is a Rectangular Prism

Volume = Cross-sectional Area x Length Surface Area depends on the type of Prism

To calculate the volume of a non-standard prism, first work out the area of its cross-section.

To calculate the surface area of a prism, it can help to draw its Net and work out the area of each part of that.

Calculate the volume of the prism below

First, calculate the area of the cross section. It's a trapezium so its area is: 5(4 + 7)/2 = 27.5 cm²

So, the volume of the prism = length x area of cross section ie. 8 x 27.5 = 220cm³

A Pyramid is a solid shape that goes up to a point at the top.

Volume = 1/3 x Area of Base x Height

Surface Area depends on the Pyramid.

A Tetrahedron is a pyramid with four faces each of which is an equilateral triangle.

Use the formula to calculate the volume of a Pyramid.

To calculate its Surface Area it often helps to draw the Net

Calculate the Surface Area of the square-based pyramid below. Give your answer in cm²

It will help to draw the net.

The area of the base is 150 x 150 = 22500 mm² The area of each triangular face is 1/2 x 150 x 125 = 9375 mm²

So, the total area is 22500 + 4 x 9375 = 60000 mm² = 600 cm²

Note that though we were told the height of the pyramid, it wasn't needed to calculate the Surface Area.

There are a number of different lines and areas associated with Circles. You need to learn which each of them is and how they relate to each other.

A Radius is a line from the Centre to the Circumference

A Diameter is a Chord which passes through the Centre of the Circle

The Circumference is the Perimeter of a Circle Circumference = 2π x Radius Circumference = π x Diameter

A Chord is a line from one part of the Circumference to another

An Arc is part of the Circumference

A Tangent is a straight line that just touches the outside of the circle. A Radius meets a Tangent at Right Angles

The area between two Radii (plural of Radius) is called a Sector (it looks like a slice of cake or what's left behind when a slice of cake has been taken).

The area between a Chord and the Circumference is a Segment

Make sure you're familiar with the various parts of a circle and how they relate to each other.

If the diameter of a circle is 8cm, what is its radius?

The diameter is twice the length of the radius so the radius is 4cm

The Area of a Circle is πr² where r is the radius

The Area of a Semi-Circle = Half the Area of a Circle The Area of a Quarter Circle = Quarter the Area of a Circle.

Be careful if you're told the Diameter of the circle rather than the Radius. Divide by half and then use the formula Area = πr²

The diameter of a semi-circular patio is 4m. What is its area to 1 d.p.?

The diameter is 4m so the radius is 2m. Area of a circle = πr² so, area of this semi-circle is: (π x 2²)/2 = 2π = 6.3 to 1 d.p. Area of patio = 6.3m² to 1 d.p.

The Circumference of a Circle is its Perimeter.

The length of the Circumference is: 2π x Radius = π x Diameter

The circular part of the Perimeter of a Semi-Circle = Half the Circumference. The circular part of the Perimeter of a Quarter-Circle = Quarter the Circumference

Find the total perimeter of a semi-circle with radius 12mm. Leave your answer in terms of π

There are two parts to the perimeter of a semi-circle. The circular part and the straight part.

The circular part is 1/2 x 2π x Radius = 12π mm

The flat part is 2 x Radius = 24mm

So, the total perimeter of the semi-circle is: (12π + 24)mm = 12(π + 2)mm

An Arc is a section of the circumference of a circle. A Sector is a portion of a circle formed by two radii and the arc that joins their ends.

Let a be the angle subtended by an arc at the centre of the circle and r be the radius of the circle

The Length of an Arc is given by the formula: Arc Length = a/360° x 2πr

The Area of a Sector is given by the formula: Area = a/360° x πr²

Just apply the formulas.

If the arc of the circle denoted by the blue dotted curved line is 5π cm, and that denoted by the red dotted curved line is π cm, in terms of π, what's the area of the green sector?

Let the radius of the circle be r and the angle inside the yellow sector be a.

So, for the shorter arc, a/360 x 2πr = π And for the longer arc, (360 - a)/360 x 2πr = 5π

Multiplying both equations by 360 and dividing by π we get: 2ar = 360 360(2r) - 2ar = 5 x 360

Adding them together we get 360 x 2r = 6 x 360 So, r = 3 cm But, 2ar = 360 so, a = 60°

Therefore, the area of the green sector is 300/360 x πr² = 5/6 x 9π = 7.5π cm²

A Cylinder is a prism with a circular cross section.

Volume of a Cylinder is πr²h Curved Surface Area of a Cylinder is 2πrh Total Surface Area of a Cylinder is 2πr(h + r) where r is the radius of the cross-section and h the height.

Just apply the formulas. With a question about surface area, make sure you include/exclude the ends if appropriate.

A cylinder's base has radius 5 cm and its volume is 200π cm³. What is its height?

Volume = πr²h So, 200π = π5²h That makes h = 200/25 = 8cm

A Cone is basically a pyramid with a circular base.

Volume of a Cone is 1/3 πr²h Curved Surface Area of a Cone is πrl where r is the radius of the base, h the height and l the length of the sloping side.

You might have to use Pythagoras' theorem to work out the height or length of the sloping side

What's the volume of a cone which has a base of radius 20mm and a height of 30mm? Leave your answer in terms of π

Volume = πr²h/3 = π x 400 x 30/3 = 4000 π mm³

A Sphere is a round solid figure.

Volume of a Sphere = 4πr³/3 Surface Area of a Sphere = 4πr² where r is the radius of the sphere.

Look out for hemi-spheres (half the volume, half the surface area of a full sphere) and other fractions of spheres.

A sphere has a diameter of 30cm. To 1 d.p., what is its volume in litres?

The diameter is 30cm so the radius is 15cm Volume = 4πr³/3 ie. 4π x (15)³/3 = 14137.17 cm³ There are 1000cm³ in 1 litre so, the sphere's volume in litres is: 14.1 litres to 1 d.p.

Compound Solids are 3 dimensional figures that are made up of two or more simple figures. For example, a hemisphere attached to the bottom of a cone.

The key thing in calculating volumes and surface areas of complex figures is to break them down into their constituent parts. Be careful when calculating surface areas as some will be lost where two shapes are joined together.

A toy wooden tower consists of a square-based pyramid, height 6cm, base 6cm x 6cm on top of a cuboid, base 6cm x 6cm, height 9cm. What's the volume of the figure?

Break the shape down into its consitutent parts.

Volume of pyramid = Base x Height/3 = 6 x 6 x 6 / 3 = 72 cm³

Volume of cuboid = Length x Width x Height = 6 x 6 x 9 = 324cm³

So, the total volume is 72 + 324 = 396cm³

A frustum is the shape left when the top of a cone or pyramid is cut off by a plane parallel to the base.

It looks like a cone or pyramid with the pointy top removed, leaving a flat top and a larger flat bottom.

Volume of a cone:

Volume of a frustum = Volume of big cone − Volume of small cone (that was removed)

There is also a direct formula for the volume of a frustum:

where R is the base radius, r is the top radius, and h is the height of the frustum.

To find the volume of a frustum:1. Work out the dimensions of the full (big) cone using similar triangles if needed2. Calculate the volume of the big cone3. Calculate the volume of the small cone that was removed4. Subtract: Volume of frustum = Big cone − Small cone

A cone has base radius 6 cm and height 10 cm. A smaller cone of radius 3 cm and height 5 cm is removed from the top. Find the volume of the frustum to 1 decimal place.

Volume of big cone = ⅓ × π × 6² × 10 = 120π

Volume of small cone = ⅓ × π × 3² × 5 = 15π

Volume of frustum = 120π − 15π = 105π = 329.9 cm³ (1 d.p.)

If you're given the frustum dimensions but not the full cone, use similar triangles to find the height and radius of the complete cone.

For surface area questions, the curved surface area of the frustum = curved surface area of big cone − curved surface area of small cone. Don't forget to add the two circular ends.