Similarity, 3D Shapes and Coordinate Geometry

Key vocabulary: a face is a flat surface; an edge is where two faces meet; a vertex (plural: vertices) is a corner.

Solid	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Triangular prism	5	9	6
Square-based pyramid	5	8	5
Cylinder	3	2	0
Cone	2	1	1
Sphere	1	0	0

(Extra context — Euler's formula $F+V-E=2$ is not required by Edexcel 1MA1, but is a useful self-check for polyhedra.)

Euler's formula: $F+V-E=2$ for any polyhedron.

Cube: $6+8-12=2$ ✓. Triangular prism: $5+6-9=2$ ✓

Cross-sections: the cross-section of a prism is constant along its length. A cylinder's cross-section is a circle; a triangular prism's is a triangle.

A plan is the view from directly above. A front elevation is the view from the front. A side elevation is the view from one side.

Key rules:

• Hidden edges appear as dashed lines.
• The plan shows the full width and depth.
• Elevations show height.
• A cylinder: plan is a circle; front and side elevations are rectangles.
• A cone: plan is a circle with a dot at the centre; front elevation is a triangle.

Worked example — a square-based pyramid with base 6 cm and apex directly above the centre:

• Plan: square with diagonals drawn to show the apex position
• Front elevation: isosceles triangle
• Side elevation: isosceles triangle (identical to front if the base is square)

Distance between two points $(x_1,y_1)$ and $(x_2,y_2)$:

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Worked example — find the distance between $(-3,2)$ and $(5,-4)$:

$d=\sqrt{(5-(-3){)}^2+(-4-2{)}^2}=\sqrt{64+36}=\sqrt{100}=10$ ✓

Checking if a triangle is right-angled: compute all three side lengths and test Pythagoras.

Worked example — vertices $A(0,0)$, $B(4,0)$, $C(4,3)$:

$AB=4$, $BC=3$, $AC=\sqrt{16+9}=5$. Since $3^2+4^2=5^2$, triangle $ABC$ is right-angled at $B$. ✓

Area on a coordinate grid — use the grid or the triangle formula $A=\frac{1}{2}bh$ with the base and perpendicular height read from the coordinates.

Congruent shapes are identical in shape and size — one can be mapped exactly onto the other by rotation, reflection, or translation. Corresponding sides are equal; corresponding angles are equal; scale factor $=1$.

Similar shapes have the same angles and proportional sides — they are enlargements of each other. Scale factor $k$ may be any positive value; when $k=1$ the shapes are also congruent.

Property	Congruent	Similar
Same angles	Yes	Yes
Same side lengths	Yes	Proportional (ratio $k$)
Same area	Yes	Area ratio $k^2$

Identifying similar triangles: two triangles are similar if they share two equal angles (AA — the third angle is then forced equal by the angle sum).

Worked example — triangles $PQR$ and $XYZ$ have $\angle P=\angle X=50°$ and $\angle Q=\angle Y=70°$. Show they are similar and find $XZ$ if $PQ=6$, $XY=9$, $PR=8$.

AA criterion: two pairs of equal angles → similar. $k=XY/PQ=9/6=1.5$

$XZ=PR\times k=8\times 1.5=12$ cm ✓

Two shapes are similar if one is an enlargement of the other (same angles, proportional sides). The ratio of corresponding sides is the linear scale factor $k$.

Finding a missing length in similar triangles:

1. Identify the corresponding sides.
2. Find the scale factor: $k=\frac{\text{image side}}{\text{object side}}$.
3. Multiply (or divide) to find the unknown.

Worked example — triangles $ABC$ and $DEF$ are similar. $AB=5$ cm, $DE=8$ cm, $BC=10$ cm. Find $EF$.

$k=8/5=1.6$. $EF=10\times 1.6=16$ cm ✓

Worked example — two similar trapezoids have corresponding sides $5$ cm and $12$ cm. The smaller has perimeter 28 cm. Find the larger perimeter.

$k=12/5=2.4$. Larger perimeter $=28\times 2.4=67.2$ cm ✓

If the linear scale factor between two similar shapes is $k$:

$\text{Area scale factor}=k^2\text{Volume scale factor}=k^3$

Worked example — two similar cones have heights 4 cm and 10 cm. The smaller has surface area 48 cm² and volume 32 cm³. Find the surface area and volume of the larger.

$k=10/4=2.5$

Surface area: $48\times 2.5^2=48\times 6.25=300$ cm²

Volume: $32\times 2.5^3=32\times 15.625=500$ cm³ ✓

Reverse problem — two similar cuboids have volumes 54 cm³ and 128 cm³. Find the ratio of their surface areas.

$k^3=128/54=64/27⟹k=\frac{4}{3}$

Area ratio $=k^2=\frac{16}{9}$ ✓

1. Plans and elevations — showing a solid edge as dashed

Visible edges are drawn as solid lines; only hidden (internal) edges use dashed lines. Confusing these misrepresents the shape.

2. Similar shapes — using the area scale factor to find missing lengths

Missing lengths scale by $k$, not $k^2$. If the area scale factor is 4, the linear scale factor is $\sqrt{4}=2$.

3. Distance formula — forgetting to square root at the end

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$ — the result of the Pythagorean sum is still under the square root.

4. Volume scale — cubing the wrong thing

If the height ratio is $3:5$, the volume ratio is $3^3:5^3=27:125$. Squaring instead (giving $9:25$) gives the area ratio.

Mistake	Correction
"Scale factor 2, so area scales by 2"	Area scales by $2^2=4$
"A cuboid has 8 faces"	A cuboid has 6 faces (top, bottom, and 4 sides)
"Distance $(-1,3)$ to $(2,7)$: $3+4=7$"	$d=\sqrt{3^2+4^2}=\sqrt{25}=5$