GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Circumference, area of circle and 3D solids

Here's a strange fact: unroll the sloping side of a cone and you don't get a triangle. You get a slice of a bigger circle.

Maths · Cones

Slice a cone down the middle

h (perpendicular height) 8.0 cm. r (base radius) 6.0 cm. l (slant length) 10.0 cm. angle at O 90°. Relationship: l² = h² + r², so l is always longer than h

h (perpendicular height)8.0 cmr (base radius)6.0 cml (slant length)10.0 cmangle at O90°Drag the apex up and down

l² = h² + r², so l is always longer than h

Drag the apex up and down. Watch which lengths change, and which one never does.

Exam line: The curved surface needs the slant length l. The volume needs the perpendicular height h.

Watch out: l slopes, h stands straight up. Putting l into a volume formula gives an answer that is too big.

Worked example · a closed cone

Problem

A closed right cone has base radius 5 cm and perpendicular height 12 cm. Find its total surface area and its volume. Keep π in your working, then round at the end.

Maths · Spheres

Where did this answer go wrong?

A solid hemisphere has radius 6 cm. Find its volume in terms of π.

A student's answer — which line goes wrong?

The sphere formula is V = (4/3)πr³, and a hemisphere is half of that.
Put r = 6 in: (4/3) × 6 = 8, then cube the 8 to get 8³ = 512.
So the sphere's volume is 512π cm³.
Halve it for the hemisphere: 256π cm³.

Exam line: Cube the radius first. Only r gets the power.

Maths · Spheres

Work backwards from the surface area

A sphere has surface area 400π cm². Find its radius.

The surface area of a sphere is A = 4πr².
Put in the area we know: 4πr² = 400π.
missing step

Which line is step 3?

Divide both sides by 4π: r² = 100.Divide both sides by 4: r² = 100π.Multiply both sides by 4π: r² = 1600π².

Exam line: Undo the formula with inverse operations, one at a time.

Maths · Displacement

What is the stone's volume?

A measuring cylinder holds water up to the 50 ml mark. A small irregular stone is lowered in, and the water level rises to the 68 ml mark.

Which is closest to what you think the stone's volume is?

The stone's volume is the rise in water level: 68 − 50 = 18 ml, and 18 ml takes up the same space as 18 cm³.The stone's volume is 68, because that's what the scale says at the end.Displacement gives the stone's volume exactly, however many decimal places you like.18 ml is a measure of liquid, so it can't be written as a volume in cm³.

How sure are you?

I'm sureI think soI'm guessing

The stone's volume is the rise in water level: 68 − 50 = 18 ml, and 18 ml takes up the same space as 18 cm³.

This matches the evidence

Right. The water level rose by the amount of space the stone takes up. One millilitre occupies the same space as one cubic centimetre, so the stone is 18 cm³.

The stone's volume is 68, because that's what the scale says at the end.

This is a common misconception

The scale reading at the end includes the water that was already there. The reading is not the volume of the object. Only the change in the level, 68 − 50, belongs to the stone.

Displacement gives the stone's volume exactly, however many decimal places you like.

This is a common misconception

The method is sound, but you can only read the water level as precisely as the scale allows. Displacement has a degree of inaccuracy limited by the scale, so the answer is a good estimate rather than an exact value.

18 ml is a measure of liquid, so it can't be written as a volume in cm³.

This is a common misconception

Millilitres and cubic centimetres measure the same thing here. One millilitre occupies the same space as one cubic centimetre, so 18 ml of displaced water is a space of 18 cm³.

Subtract the starting level from the final level, treat 1 ml as 1 cm³, and remember the scale limits how accurate the answer can be.

Exam line: Volume is the change in water level, not the final reading.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Spot the right length and every formula in this topic gets easier.

What you need to know

A circle's circumference is 2πr (also πd) and its area is πr². An exact answer can stay in terms of π.
Have a goDev is sure a circle of radius 5 cm has circumference π × 5². Dev is very confident. Dev is also wrong. What should it be?
2 × π × 5 = 10π cm. Dev has used the area formula.
π r² gives area, so it only appears when you're finding space inside. Circumference is a length, so r appears once: 2πr (or πd).
A sphere has volume (4/3)πr³ and surface area 4πr². Cube the radius first, then multiply by the constants.
A hemisphere is half a sphere. Its volume is half the sphere's, and its total surface area is 2πr² + πr² = 3πr².
To get a sphere's radius from its surface area, divide by 4π, then take the positive square root.
A cone has base radius r, perpendicular height h and slant length l. In a right cone, l² = h² + r².
Have a goA cone has base radius 8 cm and perpendicular height 15 cm. Do you add or subtract the squares to get l, and how long is it?
Add: l² = 225 + 64 = 289, so l = 17 cm.
l slopes across from the apex to the rim, so it is the longest side of the right-angled triangle. That's why l² = h² + r² adds the squares.
A cone's curved surface unrolls into a sector of radius l with arc 2πr. Its area is πrl, and a closed cone adds the base: πr² + πrl.
Cone volume is one third of the cylinder with the same base and height: (1/3)πr²h. Use h, never l.
Have a goA cone has radius 3 cm, perpendicular height 7 cm and slant length about 7.6 cm. Which length goes in (1/3)πr²h, and what is the volume in terms of π?
Use h = 7: V = (1/3) × π × 9 × 7 = 21π cm³.
Volume uses the perpendicular height h. The slant length belongs to the curved surface, and using it here would make the volume too big.
A pyramid has a polygonal base and triangular faces meeting at an apex. In a right pyramid, the apex is directly above the centre of the base.
Any pyramid has volume (1/3) × base area × perpendicular height, whatever the base shape. Rearrange it to find a missing base area or height.
Displacement finds an irregular volume: it's the change in water level, and 1 ml takes up the same space as 1 cm³.

The big picture

Every formula in this topic wants one particular length. Circles want the radius. Cones want the slant length for the curved surface and the perpendicular height for the volume. Spheres want the radius cubed before the constants. Pyramids want the perpendicular height, never an edge of a face.

Key points

1Circle: C = 2πr (= πd) and A = πr². Leave π in an exact answer.

2Sphere: V = (4/3)πr³ and A = 4πr². Cube r first. A hemisphere has half the volume, and total surface area 3πr².

3Cone: l² = h² + r². Curved surface πrl, closed cone πr² + πrl, volume (1/3)πr²h with h.

4Pyramid: V = (1/3) × base area × perpendicular height, right or oblique, any base shape.

5Displacement: volume is the change in water level, 1 ml = 1 cm³, and the scale limits accuracy.

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Worked example

Problem

A right pyramid has a square base of side 6 cm and a perpendicular height of 4 cm. Each triangular face is 5 cm tall, measured up its middle. (a) Find the volume. (b) A taller pyramid with the same base has volume 84 cm³. Find its perpendicular height.

⚠ Watch out

Using the slant length l (or an edge of a triangular face) where the perpendicular height belongs in a volume. A second trap: in (4/3)πr³, only r gets the power, so cube it before you multiply by the constants.

🧠

Memory hook

Slant for the skin, straight-up for the stuff. l goes in the curved surface, h goes in the volume.

✓

Check yourself

A cone has radius 12 cm and perpendicular height 16 cm. Which of l and h goes in the curved surface area, and which in the volume? Then find l, both answers, leaving π in.

Flashcards

(15)

Circumference and area of a circle of radius r?

C = 2πr (which is also πd). A = πr².

Volume and surface area of a sphere?

V = (4/3)πr³ and A = 4πr².

In V = (4/3)πr³, what do you do to r first?

Cube it. Only r gets the power, then multiply by (4/3)π.

Volume of a hemisphere of radius r?

Half the sphere's: (2/3)πr³.

Why is a hemisphere's total surface area 3πr² and not 2πr²?

The curved part is half of 4πr², which is 2πr². The flat circular face adds πr².

How do you find a sphere's radius from its surface area?

Divide by 4π, then take the positive square root, because r is a length.

On a cone, what are r, h and l, and how are they linked?

Base radius, perpendicular height, slant length. In a right cone l² = h² + r², so l is the longest.

What does a cone's curved surface unroll into, and what is its area?

A sector of radius l with arc length 2πr. Its area is πrl.

Surface area of a closed cone?

πr² + πrl: the circular base plus the curved surface.

Volume of a cone, and which height does it use?

(1/3)πr²h, one third of the cylinder with the same base and height. It uses the perpendicular height h, not l.

Right pyramid or oblique pyramid?

Right: the apex is directly above the centre of the base. Oblique: it isn't.

Volume of any pyramid?

(1/3) × base area × perpendicular height, whatever the shape of the base.

Which length does a pyramid's volume use: an edge of a triangular face, or the perpendicular height?

The perpendicular height, even for an oblique pyramid. A face's edge is longer and gives a volume that is too big.

How do you find the volume of an irregular object by displacement?

Put it in water. Its volume is the change in water level. 1 ml = 1 cm³, and the scale limits the accuracy.

What does an exact answer look like for a circular shape?

It keeps π in it, like 100π cm³. Round only at the end, after finding one third or any other fraction.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Circumference, area of circle and 3D solids

Where did this answer go wrong?

Work backwards from the surface area

Which solid has which feature?

What is the stone's volume?

More Edexcel GCSE Maths topics