Meditaliano IMAT Preparation

Mathematics: Lesson 6 - Plane & Solid Geometry

Lesson 6: Plane & Solid Geometry

This lesson explores both plane (2D) and solid (3D) geometry. We will begin with 2D plane geometry—focusing on triangles, quadrilaterals, polygons, and circle theorems (inscribed angles, chords, and tangents)—and then transition into 3D solid geometry, including volume, surface area, and coordinates. This comprehensive review covers high-yield topics frequently tested in the IMAT.

Learning Objectives (LO M4.1):

  • Apply 2D geometry formulas for triangles, quadrilaterals, and regular polygons.
  • Solve circle geometry problems using the Inscribed Angle Theorem, Thales's Theorem, and alternate segments.
  • Accurately calculate the volume and surface area of common 3D shapes (Cubes, Prisms, Cylinders, Cones, Spheres).
  • Understand and apply Euler's Formula for polyhedra.
  • Master 3D Coordinate Geometry (Distance, Midpoint).
  • Solve Inscribed and Circumscribed solid problems (e.g., Sphere inside a Cube).
  • Master the relationship between scale factors for length, area, and volume (The Square-Cube Law).
  • Advanced: Calculating properties of the Regular Tetrahedron.
  • Advanced: Analyzing Cross-Sections and Solids of Revolution.

Part 1: Plane Geometry & Circle Theorems

Plane geometry is the study of flat, two-dimensional shapes. The IMAT frequently tests relationships in triangles, properties of quadrilaterals and polygons, and circle theorems.

1.1 Triangles & Similarity

Triangles are the most fundamental polygon. We describe their area and scaling properties using both classical and trigonometric methods.

Triangle Area Formulas

  • Standard Area: $$ A = \frac{1}{2} b h $$ (where $b$ = base, $h$ = height)
  • Trigonometric Area: $$ A = \frac{1}{2} a b \sin(C) $$ (where $a, b$ are sides, and $C$ is the included angle)
  • Heron's Formula: $$ A = \sqrt{s(s-a)(s-b)(s-c)} $$ (where $s$ is the semi-perimeter: $s = \frac{a+b+c}{2}$)
h b (base) a c C

Triangle Similarity

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. If the ratio of corresponding side lengths is $1:k$:

Length ratio = $1 : k \quad \Rightarrow \quad$ Area ratio = $1 : k^2$

Special Triangles & Pythagorean Triples

Memorizing special right-angled triangles saves valuable time on the IMAT:

  • 30°-60°-90° Triangle: Side ratio is $1 : \sqrt{3} : 2$ (the hypotenuse is double the shortest side).
  • 45°-45°-90° Triangle: Side ratio is $1 : 1 : \sqrt{2}$ (isosceles right triangle).
  • Pythagorean Triples: Whole number sets that satisfy $a^2 + b^2 = c^2$, such as (3, 4, 5), (5, 12, 13), (7, 24, 25), and (8, 15, 17). Any integer multiples (e.g., 6, 8, 10) are also triples.

1.2 Polygons & Quadrilaterals

Quadrilaterals and other polygons have specific properties that dictate their interior angles and area calculations.

Polygon Angle Formulas

  • Sum of Interior Angles: $$ S_{\text{int}} = (n-2) \times 180^\circ $$ (where $n$ is the number of sides)
  • Sum of Exterior Angles: $$ S_{\text{ext}} = 360^\circ $$ (always true for any convex polygon, regardless of $n$)

Quadrilateral Area Formulas

  • Parallelogram: $A = b \times h$ (base times perpendicular height).
  • Trapezium (Trapezoid): $A = \frac{a+b}{2} \times h$ (average of parallel bases times height).
  • Rhombus & Kite: $A = \frac{1}{2} d_1 d_2$ (half the product of the perpendicular diagonals).
h a b

Trapezium Area: $A = \frac{a+b}{2}h$

d₁ d₂

Rhombus Area: $A = \frac{1}{2}d_1 d_2$

1.3 Circles & Sector Geometry

A circle's properties are dictated by its radius $r$. Often, the IMAT asks for the length of a specific arc or the area of a sector.

Circle & Sector Formulas

Using Degrees ($\theta$)

  • $$ \text{Circumference} = 2\pi r $$
  • $$ \text{Circle Area} = \pi r^2 $$
  • $$ \text{Arc Length } s = 2\pi r \left(\frac{\theta}{360^\circ}\right) $$
  • $$ \text{Sector Area } A = \pi r^2 \left(\frac{\theta}{360^\circ}\right) $$

Using Radians ($\theta$)

  • $$ \text{Arc Length } s = r\theta $$
  • $$ \text{Sector Area } A = \frac{1}{2} r^2 \theta $$
r θ s (arc)

1.4 Circle Theorems & Angle Relationships

Circle theorems represent visual puzzles that can be solved quickly if you know the geometric rules.

Fundamental Circle Theorems

Inscribed Angle Theorem

The angle subtended by an arc at the center is exactly **twice** the angle subtended at any point on the circumference.

$$ \theta_{\text{center}} = 2 \theta_{\text{circumference}} $$

Corollary (Thales's Theorem): Any angle inscribed in a semicircle is a right angle ($90^\circ$).

Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle. Its **opposite angles** are always supplementary:

$$ \alpha + \beta = 180^\circ $$

Tangent-Radius Theorem

A tangent line is perpendicular ($90^\circ$) to the circle's radius at the point of contact.

Intersecting Chords Theorem

If two chords $AB$ and $CD$ intersect inside a circle at point $P$, the products of their segments are equal:

$$ AP \cdot PB = CP \cdot PD $$

Alternate Segment Theorem

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

θ 2θ

Inscribed Angle Theorem

α β

Cyclic Quadrilateral: $\alpha + \beta = 180^\circ$

Part 2: Properties of Common 3D Shapes

Solid geometry deals with objects in three-dimensional space. The fundamental components of many solids are faces (the flat surfaces), edges (where two faces meet), and vertices (the corners where edges meet).

Euler's Formula for Polyhedra

For any convex polyhedron:

$$ F + V = E + 2 $$

Where $F$ = Faces, $V$ = Vertices, $E$ = Edges.

Check for a Cube:

Faces ($F$) = 6

Vertices ($V$) = 8

Edges ($E$) = 12

$$ 6 + 8 = 12 + 2 \implies 14 = 14 \quad \text{(Verified)} $$

The Space Diagonal

A crucial concept for cubes and cuboids (rectangular prisms) is the Space Diagonal. This is the line connecting two opposite vertices passing through the center of the shape.

d
Space Diagonal of a Rectangular Prism ($l, w, h$):
$$ d = \sqrt{l^2 + w^2 + h^2} $$
Space Diagonal of a Cube (side $a$):
$$ d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} $$
IMAT Challenge

Challenge an IMAT Question!

Official Paper: IMAT Extra

What is the volume of a cube whose total surface area is 54 cm²?

The Platonic Solids

There are exactly five regular polyhedra (faces are congruent regular polygons):

  • Tetrahedron: 4 triangular faces.
  • Cube (Hexahedron): 6 square faces.
  • Octahedron: 8 triangular faces.
  • Dodecahedron: 12 pentagonal faces.
  • Icosahedron: 20 triangular faces.

Cube

6 square faces, 12 equal edges, 8 vertices.

Cylinder

Two circular bases and one curved surface.

Cone

One circular base, one curved surface, and one apex.

Sphere

A set of points equidistant from a center point.

Rectangular Prism

6 rectangular faces, 12 edges, 8 vertices.

Square Pyramid

A polygonal base and triangular faces meeting at an apex.

Part 2.5: The Regular Tetrahedron

The regular tetrahedron is a frequently tested shape in advanced math sections. It is composed of 4 equilateral triangles. If the side length of the tetrahedron is $s$, we can derive its height, area, and volume.

h s

Height ($h$):

$$ h = s \frac{\sqrt{6}}{3} $$

Total Surface Area ($SA$):

$$ SA = s^2 \sqrt{3} $$

Volume ($V$):

$$ V = \frac{s^3 \sqrt{2}}{12} $$
Derivation Tip: The height falls on the centroid of the equilateral base. The distance from a vertex to the centroid of an equilateral triangle with side $s$ is $s/\sqrt{3}$. Using Pythagoras with the slant edge $s$: $h^2 = s^2 - (s/\sqrt{3})^2$.

Part 3: Surface Area Calculations

Surface area is the total area of all surfaces of a 3D object. Visualizing the net of a solid (what it looks like when unfolded) can be extremely helpful.

Cylinder

The surface area is the sum of the areas of the two circular bases and the curved rectangular side.

$h$ $r$
$SA = \underbrace{2\pi r^2}_{\text{Top \& Bottom Circles}} + \underbrace{2\pi r h}_{\text{Curved Side}}$

The curved side unfolds into a rectangle with height $h$ and width equal to the circumference of the base, $2\pi r$.

Cone

The surface area is the sum of the circular base and the curved sector. It requires the slant height ($l$).

$h$ $r$ $l$
$l = \sqrt{r^2 + h^2}$
$SA = \underbrace{\pi r^2}_{\text{Base Circle}} + \underbrace{\pi r l}_{\text{Curved Surface}}$

The slant height $l$ is the hypotenuse of a right-angled triangle formed by the radius $r$ and the height $h$.

Sphere

The formula for the surface area of a sphere is a fundamental result in geometry.

$SA = 4\pi r^2$

Note for IMAT: If you are asked for the surface area of a hemisphere, be careful!

  • Curved Surface Area = $2\pi r^2$ (half of a sphere)
  • Total Surface Area (including the flat base) = $2\pi r^2 + \pi r^2 = 3\pi r^2$
IMAT Challenge

Challenge an IMAT Question!

Official Paper: 2014 - Q57

Four individual spheres have radii: $r$, $1/2 r$, $2r$ and $3r$. What is the sum of their surface areas?

Part 4: Volume Calculations

Volume measures the amount of three-dimensional space an object occupies. For many shapes, it can be calculated as the area of the base multiplied by the height.

Prisms (Cubes & Cylinders)

A prism is a solid with two parallel and congruent bases. Its volume is straightforward.

$V = \text{Area of Base} \times \text{Height}$
For a Cylinder: $V = (\pi r^2)h$

Pyramids & Cones

A key insight is that the volume of a pyramid or cone is exactly one-third the volume of a prism or cylinder with the same base and height.

A cone's volume is 1/3 that of its containing cylinder.

$V = \frac{1}{3} \times \text{Area of Base} \times \text{Height}$
For a Cone: $V = \frac{1}{3}\pi r^2 h$

Sphere

The volume of a sphere is derived using advanced calculus (integration), but the formula is simple to apply.

$V = \frac{4}{3}\pi r^3$

Application: Solids of Revolution

Often in physics or advanced math questions, you are asked to imagine rotating a 2D shape around an axis. This creates a solid of revolution.

Rectangle $\to$ Cylinder

Rotating a rectangle around one side creates a cylinder.

Triangle $\to$ Cone

Rotating a right triangle around a leg creates a cone.

Semicircle $\to$ Sphere

Rotating a semicircle around its diameter creates a sphere.

Application: Density

Density relates mass and volume. This is a common application in IMAT physics and math questions.

$$ \rho = \frac{m}{V} $$

Where $\rho$ is density, $m$ is mass, and $V$ is volume.

Example: A gold cube has a side length of 2 cm. The density of gold is 19.3 g/cm³. What is the mass?

  1. Calculate Volume: $V = 2^3 = 8 \text{ cm}^3$.
  2. Rearrange Formula: $m = \rho \times V$.
  3. Calculate Mass: $m = 19.3 \times 8 = 154.4 \text{ g}$.
IMAT Challenge

Challenge an IMAT Question!

Official Paper: 2024 - Q53

Given a cylinder with a base radius of 5 cm and a height of 7 cm, what is its volume?

Part 5: Ratio of Similarity (The Square-Cube Law)

When two 3D shapes are similar, it means they are the same shape but different sizes. All corresponding lengths (like radius, height, side length) have the same ratio, which we'll call $k$.

The relationship between the length ratio, surface area ratio, and volume ratio is crucial and follows a clear pattern.

Side = 1

Side = 2

MeasurementRatioExample (Side ratio 1:2)
Length Ratio$1 : k$$1 : 2$
Surface Area Ratio$1^2 : k^2 \implies 1 : k^2$$6(1)^2 : 6(2)^2 \implies 6:24 \implies 1:4$
Volume Ratio$1^3 : k^3 \implies 1 : k^3$$(1)^3 : (2)^3 \implies 1:8$

Biology Connection: Surface Area to Volume Ratio

As a cell grows larger, its volume (needs for metabolism) increases much faster ($r^3$) than its surface area (supply capacity, $r^2$). This is why cells are small: they need a high SA:V ratio to survive. Large animals (Volume) lose heat (Surface Area) slower than small animals.

Part 6: Inscribed & Circumscribed Solids (IMAT High Yield)

A common trick in competitive exams involves placing one shape inside another. The key is to find the relationship between their dimensions.

1. Sphere Inscribed in a Cube

The sphere touches the faces of the cube.

  • The diameter of the sphere equals the side of the cube.
  • $2r = s$ or $r = s/2$
2. Cube Inscribed in a Sphere

The vertices of the cube touch the sphere.

  • The Space Diagonal of the cube equals the diameter of the sphere.
  • $s\sqrt{3} = 2r$
3. Cylinder Inscribed in a Sphere

The rims of the cylinder touch the sphere.

  • Use Pythagorean theorem with the sphere's radius ($R$), cylinder's radius ($r$), and half-height ($h/2$).
  • $R^2 = r^2 + (h/2)^2$
4. Sphere Inscribed in a Cylinder

The sphere touches the top, bottom, and sides.

  • Sphere radius = Cylinder radius ($r$).
  • Cylinder height = Sphere diameter ($2r$).
  • Volume Ratio Sphere:Cylinder is always 2:3.

Part 7: Advanced Topics & Applications

Frustums: Sliced Cones and Pyramids

A frustum is the portion of a cone or pyramid that remains after its top is cut off by a plane parallel to the base.

$r$ $R$ $h$ $l$

The volume can be found by subtracting the small, removed cone from the original, larger cone.

Strategy: Similar Triangles
If you don't know the height of the "missing" top cone, use similar triangles: $\frac{R}{r} = \frac{H_{total}}{H_{top}}$.
Volume of a Cone's Frustum:
$V = \frac{1}{3}\pi h(R^2 + Rr + r^2)$

Surface Area of a Cone's Frustum:
$SA = \pi(R+r)l + \pi R^2 + \pi r^2$
(where $l$ is the slant height of the frustum)

Composite Solids: Combining and Subtracting Shapes

Many objects are combinations of simpler shapes. To analyze them, we break them down into their basic components.

  • Volume: Simply add or subtract the volumes of the component parts.
  • Surface Area: This is more complex. Add the exposed surface areas of each part, but be sure to subtract any overlapping areas that are no longer on the surface.
Ice Cream Cone Example (Cone + Hemisphere)

$V_{total} = V_{cone} + V_{hemisphere}$

$SA_{total} = SA_{lateral\_cone} + SA_{hemisphere}$

(The circular base is internal and not part of the surface area)

Drilled Cube Example (Cube - Cylinder)

$V_{total} = V_{cube} - V_{cylinder}$

$SA_{total} = SA_{cube} - 2 \times A_{circle} + SA_{lateral\_cylinder}$

(Subtract the holes, but add the new inner wall)

Cavalieri's Principle

This is a powerful, non-calculus method for determining volume. It states: If two solids have the same height and the same cross-sectional area at every level parallel to their bases, then they have the same volume.

Same Volume

A regular stack and a skewed stack of coins have the same volume.

Part 7.5: Cross-Sections of Solids

A cross-section is the intersection of a solid body and a plane. Identifying the 2D shape of a cross-section is important for spatial reasoning questions.

Solid Cutting Plane Resulting 2D Shape
Cylinder Parallel to Base Circle
Cylinder Parallel to Axis (Vertical) Rectangle
Cube Through 3 Corners Triangle
Cube Parallel to Face Square
Sphere Any Plane Circle

Part 8: 3D Coordinate Geometry

Just like in 2D $(x, y)$, points in 3D space are defined by $(x, y, z)$. The formulas for distance and midpoint are natural extensions of the 2D versions.

Distance Formula

The distance between point $A(x_1, y_1, z_1)$ and point $B(x_2, y_2, z_2)$ is:

$$ D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} $$
Midpoint Formula

The midpoint $M$ between two points is the average of their coordinates:

$$ M = \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) $$

Part 9: Practice Problems

1. (Volume of Cube) What is the volume of a cube with a side length of 5 cm?

2. (Surface Area of Cylinder) Find the total surface area of a cylinder with a radius of 3 cm and a height of 10 cm.

3. (Volume of Cone) A cone has a radius of 6 cm and a height of 8 cm. What is its volume?

4. (Surface Area of Sphere) What is the surface area of a sphere with a radius of 10 cm?

5. (Similarity Ratio) Two spheres are similar. The ratio of their radii is 1:3. What is the ratio of their volumes?

6. (Slant Height) A cone has a height of 4 cm and a radius of 3 cm. What is its slant height?

7. (Volume of Sphere) What is the volume of a sphere with a radius of 3 cm?

8. (Surface Area of Cube) A cube has a surface area of 24 cm². What is the length of one side?

9. (Similarity Ratio) Two similar cylinders have a height ratio of 2:5. What is the ratio of their surface areas?

10. (Volume of Rectangular Prism) A prism has a length of 10 cm, a width of 4 cm, and a height of 5 cm. Find its volume.

11. (Surface Area of Cone) A cone has a radius of 5 cm and a slant height of 13 cm. What is its total surface area?

12. (Conceptual) If you double the radius of a sphere, what happens to its volume?

13. (Volume of Pyramid) A square pyramid has a base side length of 6 cm and a height of 10 cm. Find its volume.

14. (Conceptual) Which formula gives the volume of a cone?

15. (Working Backwards) The volume of a cylinder is $128\pi$ cm³. If its height is 8 cm, what is its radius?

16. (Conceptual) A cone and a cylinder have the same radius and height. The volume of the cone is 30 cm³. What is the volume of the cylinder?

17. (Surface Area of Prism) What is the surface area of a rectangular prism with dimensions 2 cm, 3 cm, and 4 cm?

18. (Similarity) The surface area ratio of two similar cones is 16:25. What is the ratio of their heights?

19. (Conceptual) If you double all the dimensions of a rectangular prism, what happens to its surface area?

20. (Working Backwards) A sphere has a surface area of $144\pi$ cm². What is its radius?

21. (Frustum Volume) A frustum of a cone has a height of 6 cm, a bottom radius of 4 cm, and a top radius of 2 cm. What is its volume?

22. (Composite Solid Volume) A solid is formed by a hemisphere of radius 3 cm on top of a cylinder with the same radius and a height of 5 cm. What is the total volume?

23. (Composite Solid Surface Area) For the solid in the previous question (hemisphere on cylinder, r=3, h=5), what is the total surface area?

24. (Frustum Surface Area) A frustum of a cone has a slant height of 5 cm, a bottom radius (R) of 6 cm, and a top radius (r) of 3 cm. Find its total surface area.

25. (Tetrahedron Height) What is the height of a regular tetrahedron with side length 6?

26. (Tetrahedron Volume) Find the volume of a regular tetrahedron with side length 2.

27. (Cross-Sections) If you cut a cylinder parallel to its base, what 2D shape do you get?

28. (Rotation) If you rotate a right triangle around its hypotenuse, what shape is formed?