GeometryHigh school4 topic explainers

Geometry course

Triangles, similarity, right-triangle trigonometry, circles, area, volume and coordinate geometry.

Topics in this course

1.Triangles
Every triangle's interior angles add to $180^\circ$ , and the longest side is always opposite the largest angle. For right triangles the Pythagorean theorem $a^2 + b^2 = c^2$ links the three sides, where $c$ is the hypotenuse. Its converse is just as useful: if the squares of the sides satisfy the equation, the triangle has a right angle.
- Triangle inequality: each side is shorter than the sum of the other two.
- Area $= \frac{1}{2} \times \text{base} \times \text{height}$ , with the height perpendicular to the chosen base.
- Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 and their multiples.
Quick check: Find the hypotenuse of a right triangle with legs 9 and 12.
$15$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study triangles
2.Circles
A circle is the set of points at a fixed distance, the radius $r$ , from a centre. The circumference is $C = 2\pi r$ and the area is $A = \pi r^2$ . The constant $\pi \approx 3.14159$ is the ratio of any circle's circumference to its diameter.
- Equation of a circle with centre $(h, k)$ and radius $r$ : $(x - h)^2 + (y - k)^2 = r^2$ .
- A tangent line is perpendicular to the radius at the point of contact.
- An inscribed angle is half the central angle that subtends the same arc.
Quick check: Find the area of a circle with radius 6, to two decimals.
$113.10$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study circles
3.Area and Volume
Area measures the space a flat shape covers, in square units; volume measures the space a solid occupies, in cubic units. Many formulas come from one idea: a prism or cylinder has volume equal to base area times height, and a pyramid or cone with the same base and height holds exactly one third of that.
- Rectangle $lw$ , triangle $\frac{1}{2}bh$ , trapezoid $\frac{1}{2}(b_1 + b_2)h$ , circle $\pi r^2$ .
- Cylinder $V = \pi r^2 h$ ; cone $V = \frac{1}{3}\pi r^2 h$ ; sphere $V = \frac{4}{3}\pi r^3$ .
- Composite shapes: split into simple parts, then add or subtract.
Quick check: Find the volume of a box 4 by 5 by 6.
$120$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study area and volume
4.Coordinate Geometry
Coordinate geometry places shapes on the $xy$ -plane so that geometric questions become algebra. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , which is the Pythagorean theorem applied to the horizontal and vertical gaps.
- Distance formula for lengths, midpoint formula for centres, slope for parallel and perpendicular checks.
- A shape is a right triangle if two sides have slopes with product $-1$ .
- Perpendicular bisector: passes through the midpoint with the negative reciprocal slope.
Quick check: Find the distance between $(-2, 3)$ and $(4, 11)$ .
$10$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study coordinate geometry

Topics in this course

1.Triangles

2.Circles

3.Area and Volume

4.Coordinate Geometry

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