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International Baccalaureate Mathematics

Geometry and Trigonometry

Syllabus Content

The distance between two points in three-dimensional space, and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane

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Official Guidance, clarification and syllabus links:

In SL examinations, only right-angled trigonometry questions will be set in reference to three-dimensional shapes.

In problems related to these topics, students should be able to identify relevant right-angled triangles in three-dimensional objects and use them to find unknown lengths and angles.

Formula Booklet:

Distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\)

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

Coordinates of the midpoint of a line segment with endpoints \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\)

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

Volume of a right-pyramid

$$ V = \frac{1}{3}Ah, \text{ where } A \text{ is the area of the base, } h \text{ is the height} $$

Volume of a right cone

$$ V = \frac{1}{3}\pi r^2h, \text{ where } r \text{ is the radius, } h \text{ is the height} $$

Area of the curved surface of a cone

$$ A = \pi rl, \text{ where } r \text{ is the radius, } l \text{ is the slant height} $$

Volume of a sphere

$$ V = \frac{4}{3}\pi r^3, \text{ where } r \text{ is the radius} $$

Surface area of a sphere

$$ A = 4\pi r^2, \text{ where } r \text{ is the radius} $$

Volume of a hemisphere:

$$ V = \frac{2}{3}\pi r^3 $$

where \(r\) is the radius of the hemisphere.

Surface area of a hemisphere (including the base):

$$ A = 3\pi r^2 $$

where \(r\) is the radius of the hemisphere.

For combinations of solids, the volume and surface area are typically found by summing the respective volumes and surface areas of the individual solids, subtracting any parts that are not part of the exterior. When two solids intersect, the volume of intersection would typically be subtracted from the total.

This video on 3D shapes is from Revision Village and is aimed at students taking the IB Maths Standard level course

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