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Surface Area and Similarity of Solids: Homothety in Space

A homothety (scaling) with centre O and ratio k moves every point to k times its distance from O. Every length is multiplied by k, every surface area by k², and every volume by k³, while angles and shape stay the same. Surface area formulas: cylinder 2πr(h + r), cone πr(l + r), sphere 4πr². Cutting a cone or pyramid by a plane parallel to its base gives a smaller similar solid.

🎬 Step-by-step story

  1. The blue solid is the original. O is a point. We can draw rays from O through the corners of the solid.
  2. Now k = 2. Every point moved twice as far from O along its ray. This is a homothety. Every length doubled, and the shape did not change.
  3. Look at one face. Edge doubled, so you can fit 2 × 2 = 4 times as many small squares. Surface area grows by k².
  4. Count the small cubes inside. 2 × 2 × 2 = 8 of them fill the big cube. Volume grows by k³.
  5. A cone does the same at k = 3: surface area × 9, volume × 27. Formulas like πr(l + r) and ⅓πr²h scale in exactly this way.
  6. Your turn. Pick a shape, slide k, and read how length, area and volume change.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

What does the centre O do?

O is the fixed point. Every point of the solid is pushed along its ray from O to k times the distance. If O were somewhere else the image would sit elsewhere, but it would be the same size.

Does scaling change the angles?

No. Look at the rays and the edges: the big solid has the same corners and the same angles. Only the lengths change.

Why is area k² and not 2k?

Each tile on the face is stretched in two directions, not one. On the doubled face you can count 2 × 2 = 4 tiles.

Why is volume k³?

The solid is stretched in three directions: length, width and height. Count the small cubes: 2 × 2 × 2 = 8.

Do the cone and sphere formulas scale in the same way?

Yes. S = πr(l + r) uses two lengths, so × k². V = ⅓πr²h uses three, so × k³. Slide k at the cone and read the readout.

What if k is less than 1?

Then the image is smaller. For k = 0.5 lengths halve, areas become a quarter, volumes an eighth. Slide k to 0.5 and see.

Homothety: scaling from a point

Choose a point O (the centre) and a number k (the ratio). A homothety sends every point P to the point P′ on the ray OP with OP′ = k × OP. If k = 2, every point goes twice as far from O. If k = ½, every point moves halfway towards O. If k is negative, P′ goes to the opposite side of O.

Two things never change: angles and shape. Parallel lines stay parallel. So the image is similar to the original. Two solids are similar when one is the image of the other by a homothety, possibly followed by moving it around (turning or flipping) without changing size.

Lengths, areas and volumes under scaling

If the ratio is k (take k > 0):

Cube check: edge a → ka. Area 6a² → 6k²a². Volume a³ → k³a³. Notice that you can use ratios even when you do not know the shapes: if two similar solids have lengths in the ratio m : n, their areas are in the ratio m² : n² and their volumes in m³ : n³.

Surface area formulas of common solids

The surface area is the total area of the outside. Use a flattened net to see it.

Each formula is made of two lengths multiplied, so under scaling all of them become k² times bigger.

Cutting a cone or pyramid: a similar piece

Cut a cone or pyramid by a plane parallel to its base. The top piece is a smaller cone or pyramid similar to the whole, with centre O at the apex. If the cut is at a fraction k of the height from the apex, then the small solid has all lengths × k, surface area × k², volume × k³. For k = ½: the top has ⅛ of the volume, so the lower part (the frustum) has ⅞ of it.

Solving similarity problems

  1. Find the ratio of any two matching lengths. That is k.
  2. For area, use k². For volume, use k³.
  3. If given areas or volumes, take a square root or cube root to get k back: k = √(area ratio) = ∛(volume ratio).
  4. Check units: cm → cm² → cm³.

Careful: the ratio of volumes is not the same as the ratio of lengths. Mixing them is the commonest mistake.

Try it: sugar cubes

Take 8 small sugar cubes (or dice). Make a 2 × 2 × 2 big cube. Count the squares on one face (4) and the cubes inside (8). Now predict: for a 3 × 3 × 3 cube, how many squares on a face, and how many cubes? Check with the 3D by sliding k to 3.

Key formulas and definitions

Worked examples

1. A cube has edge 3 cm. It is scaled by k = 2. Find the new surface area and volume.

Original S = 6 × 9 = 54 cm², V = 27 cm³. New S = 54 × 4 = 216 cm². New V = 27 × 8 = 216 cm³.

2. Two similar cones have heights 6 cm and 9 cm. The small one has volume 40 cm³. Find the volume of the larger one.

k = 9/6 = 1.5. Volume × k³ = 40 × 3.375 = 135 cm³.

3. A sphere of radius 3 cm is scaled by k = 2. Find the new surface area.

New radius = 6 cm. S = 4π × 36 = 144π cm². (Check: original 36π × 4 = 144π.)

4. A cone has r = 3 cm and l = 5 cm. Find its total surface area, and the area of a similar cone with scale factor 3.

S = πr(l + r) = π × 3 × 8 = 24π cm². Similar cone: 24π × 9 = 216π cm².

5. A cone of volume 240 cm³ is cut by a plane parallel to its base halfway up the height. Find the volume of the top cone and of the frustum.

k = ½, so top cone = 240 × ⅛ = 30 cm³. Frustum = 240 − 30 = 210 cm³.

6. The volumes of two similar solids are in the ratio 8 : 27. Find the ratio of their surface areas.

k = ∛(8/27) = 2/3. Area ratio = k² = 4/9, so 4 : 9.

7. A 1 : 20 model of a statue needs 0.5 m² of paint. How much paint area does the real statue have, and how many times larger is its volume?

k = 20. Area = 0.5 × 400 = 200 m². Volume is 20³ = 8000 times larger.

Common mistakes

Practice quiz

1. A solid is scaled by k = 3. Its surface area becomes:
2. A solid is scaled by k = 3. Its volume becomes:
3. Under a homothety, which stays the same?
4. Two similar solids have volumes 1 : 8. The ratio of lengths is:
5. A cone is cut halfway up by a plane parallel to the base. The top cone is what fraction of the whole volume?

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is homothety in simple words?

It is scaling from a fixed point O: every point moves along its ray from O to k times the distance. The picture gets bigger or smaller but the shape stays the same.

Why do areas change by k² and volumes by k³?

An area is made of two lengths and a volume of three lengths. Each length grows by k, so k × k and k × k × k.

Are two cones with the same angle at the apex always similar?

Yes, if they have the same shape (the same ratio of radius to height), one is a scaled copy of the other.

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