Geometry & Shapes - Formulas
IB Mathematics Analysis & Approaches (SL & HL)
📐 2D Shapes - Area & Perimeter
Square:
\[\text{Area} = s^2\]
\[\text{Perimeter} = 4s\]
where \(s\) = side length
Rectangle:
\[\text{Area} = \ell \times w\]
\[\text{Perimeter} = 2(\ell + w)\]
where \(\ell\) = length, \(w\) = width
Triangle:
\[\text{Area} = \frac{1}{2}bh\]
\[\text{Perimeter} = a + b + c\]
where \(b\) = base, \(h\) = height, \(a, b, c\) = side lengths
Parallelogram:
\[\text{Area} = b \times h\]
where \(b\) = base, \(h\) = perpendicular height
Trapezoid (Trapezium):
\[\text{Area} = \frac{1}{2}(a + b)h\]
where \(a, b\) = parallel sides, \(h\) = perpendicular height
⭕ Circles
Area:
\[\text{Area} = \pi r^2\]
Circumference:
\[C = 2\pi r = \pi d\]
where \(r\) = radius, \(d\) = diameter
Sector Area:
\[\text{Area} = \frac{1}{2}r^2\theta\]
where \(\theta\) is in radians
Arc Length:
\[s = r\theta\]
where \(\theta\) is in radians
📦 Prisms & Cuboids
General Prism Volume:
\[V = A \times h\]
where \(A\) = area of cross-section, \(h\) = height/length
Cube:
\[V = s^3\]
\[\text{Surface Area} = 6s^2\]
where \(s\) = side length
Cuboid (Rectangular Prism):
\[V = \ell \times w \times h\]
\[\text{Surface Area} = 2(\ell w + wh + \ell h)\]
where \(\ell\) = length, \(w\) = width, \(h\) = height
🥫 Cylinders
Volume:
\[V = \pi r^2 h\]
Given in formula booklet
Curved Surface Area:
\[\text{CSA} = 2\pi rh\]
Given in formula booklet
Total Surface Area:
\[\text{TSA} = 2\pi r(r + h)\]
where \(r\) = radius, \(h\) = height
🔺 Pyramids
General Pyramid Volume:
\[V = \frac{1}{3}Ah\]
where \(A\) = base area, \(h\) = perpendicular height
Given in formula booklet
Square Pyramid:
\[V = \frac{1}{3}s^2h\]
where \(s\) = base side length, \(h\) = perpendicular height
🍦 Cones
Volume:
\[V = \frac{1}{3}\pi r^2 h\]
Given in formula booklet
Curved Surface Area:
\[\text{CSA} = \pi r \ell\]
Given in formula booklet
Total Surface Area:
\[\text{TSA} = \pi r(r + \ell)\]
Slant Height:
\[\ell = \sqrt{r^2 + h^2}\]
where \(r\) = radius, \(h\) = perpendicular height, \(\ell\) = slant height
🌐 Spheres & Hemispheres
Sphere Volume:
\[V = \frac{4}{3}\pi r^3\]
Given in formula booklet
Sphere Surface Area:
\[\text{Surface Area} = 4\pi r^2\]
Given in formula booklet
Hemisphere Volume:
\[V = \frac{2}{3}\pi r^3\]
Hemisphere Surface Area:
Curved surface only: \(2\pi r^2\)
Including base: \(3\pi r^2\)
where \(r\) = radius
📍 3D Coordinate Geometry
Distance Between Two Points in 3D:
\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]
Between points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\)
Given in formula booklet
Midpoint in 3D:
\[M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)\]
Between points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\)
Given in formula booklet
📐 Pythagorean Theorem
For Right-Angled Triangles:
\[a^2 + b^2 = c^2\]
where \(c\) = hypotenuse, \(a\) and \(b\) = other two sides
In 3D Space:
\[d^2 = \ell^2 + w^2 + h^2\]
Space diagonal of a cuboid
🔨 Composite Shapes
General Approach:
For Area/Volume: Break into simpler shapes, calculate each part, then add or subtract
For Surface Area: Identify all exposed faces, calculate each area, then sum
Common Combinations:
• Hemisphere on cylinder
• Cone on cylinder
• Sphere on cone
• Multiple prisms joined together
Important: Exclude shared faces when calculating total surface area
💡 Exam Tip: Most volume and surface area formulas for 3D shapes are given in the IB formula booklet. Always check the booklet during your exam. Remember that surface area is measured in square units (cm², m²) while volume is in cubic units (cm³, m³). For composite shapes, sketch and label all parts before calculating. Use your GDC to verify numerical answers.