Geometry Refresher Trivia Quiz

Answer: ½ × b × h

The area of a triangle is calculated using the formula ½ × base × height. This works because a triangle can be thought of as half of a parallelogram with the same base and height (see image).

Imagine a triangle, any triangle. Flip it vertically, flip it horizontally, bring it together next to its twin, and you've got yourself a parallelogram. (If you start with a right triangle, you'll end up with a special kind of parallelogram we lovingly know as a rectangle.)

Well, we know the area of a parallelogram. It's b × h. Since the parallelogram is made up of two equal triangles of equal area, we can easily deduce the area of each by taking half that value.

The base can be any side of the triangle, as long as the corresponding perpendicular height is used.

Other formulas for triangle area exist, such as Heron's formula (which uses the lengths of all three sides), but ½ × b × h is the most direct.

Example:

If a triangle has a base of 10 units and a height of 6 units, its area would be:

½ × 10 × 6 = 30 square units.

And you thought this would be hard.

Answer: 180°

The sum of the interior angles of any triangle is always 180 degrees. This is a basic property of Euclidean geometry. No matter what type of triangle you're puzzling over (scalene, isosceles, equilateral) the three interior angles will always add up to 180°.

One way to see this is by drawing a line parallel to one side of the triangle through the opposite vertex, represented in the image by the line segment AB.

The three angles (a, b, and c) are formed under the line segment. A straight angle measure 180°, so we know that a + b + c = 180°.

Why does it matter? The line segment AB creates alternate interior angles that match the lower angles of the triangle as labelled. Therefore, the angles of the triangle must also total 180°.

Answer: Pythagorean Theorem

The much-ballyhooed Pythagorean Theorem states that for a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Spelled out in words, it sounds dreadful, but you know the formula:

a² + b² = c², where c is the length of the hypotenuse.

This theorem is one of the most famous results in mathematics and has been known for thousands of years (since the time of Pythagoras, I'm told). It provides a direct relationship between the lengths of the sides of a right triangle and is used in countless applications, from architecture and navigation to physics and computer graphics.

Example:

If the legs of a right triangle measure 3 and 4 units, the hypotenuse will be:

c = √(3² + 4²) = √25 = 5 units.

The Pythagorean Theorem connects to other areas of math as well. It serves as one of the foundations of trigonometry (check out its less famous cousin, the Pythagorean trigonometric identity: sin²θ + cos²θ = 1) and the distance formula in (ugh) analytical geometry.

Answer: 2 × π × r

The formula for the circumference of a circle is 2πr, where r is the radius. This comes from the definition of π (pi) as the ratio of a circle's circumference to its diameter:

C / d = π

Multiply each side by d:

C = π × d

Since the diameter is twice the radius (d = 2r), the circumference becomes:

C = π × d = 2 × π× r.

Example:
A circle with a radius of 7 units has a circumference of:

2 × π × 7 ≈ 43.98 units.

Answer: π × r²

The formula for the area of a circle is π × r², where r is the radius. This expresses the amount of space contained within the circle.

The formula dates back thousands of years, but can be imagined more simply with calculus, by integrating the circumference relationship into two dimensions.

You can live your whole life without knowing or caring how that formula is derived, and if that's your pleasure, you can skip the rest of this question's information.

My kind of people! Let's not just forget all the hard work of those old Greeks with long names! If you take a regular polygon (equal sides and interior angles) of n sides and keep increasing n, the polygon starts to resemble a circle more and more.

So consider an equilateral triangle (n=3). Then a square (n=4). Then an octagon (n=8). And so on. As n gets very large, (100 sides, 1000 sides, 10000 sides), the edges smooth out, and it can be hard to tell the difference from a circle with the naked eye. If we imagine that the number of sides is essentially infinite, then it's not hard to imagine that it would be a literal circle we're looking at.

Well, the area of a regular polygon is:

A = ½ × apothem × perimeter

Wait, wait! Don't run away now! The apothem is just the perpendicular distance from the center to the midpoint of a side (similar to a circle's radius), and the perimeter is just the length of all the sides added together (similar to a circle's circumference).

Now, if we take the above formula and reframe the terms, we get:

A = ½ × radius × circumference

The circumference is 2 × π × r, so we get:

A = ½ × r × (2 × π × r)

Simplify, and:

A = π × r²

And a fine formula it is! Aren't you glad you read the whole thing?

Example:

A circle with a radius of 5 units has an area of:

A = π × 25 ≈ 78.54 square units.

Answer: Equal in length

In a rectangle, the diagonals are always equal in length. This follows from the fact that a rectangle is a type of parallelogram with right angles. Since opposite sides are equal and parallel, the diagonals each connect the same pair of opposite corners, making them congruent.

However, unlike in a square, the diagonals of a rectangle are not always perpendicular. They bisect each other but usually cross at an angle other than 90°.

Example:

In a 6 × 4 rectangle, both diagonals measure √(6² + 4²) = √52 ≈ 7.21 units by using the Pythagorean theorem.

Answer: 360°

The sum of the interior angles of any quadrilateral is always 360 degrees. A quadrilateral has four sides, and if you draw a diagonal as shown, you can divide it into two triangles. Since each triangle's interior angles add up to 180°, the total for both triangles is 180° + 180° = 360°.

This rule holds true for all types of quadrilaterals... squares, rectangles, parallelograms, trapezoids, rhombuses, and irregular quadrilaterals (shown).

For example, in a rectangle, each of the four angles is 90°, and 90 × 4 = 360°. In an irregular quadrilateral, the angles may differ, but they still add up to 360°.

This principle is part of a broader pattern: the sum of the interior angles of an n-sided polygon is given by (n − 2) × 180°. For quadrilaterals, n = 4, so (4 − 2) × 180° = 360°.

Answer: Equilateral triangle

A triangle with all three sides equal is called an equilateral triangle. In addition to having equal side lengths, all three of its interior angles are also equal, each measuring 60°. This makes it a highly symmetrical shape and a special case of both isosceles (at least two equal sides) and acute triangles (all angles less than 90°).

Equilateral triangles have unique geometric properties. For example, their medians, altitudes, and angle bisectors all coincide at the same point (the centroid, which is also the incenter and circumcenter).

Answer: Cuboid

A cuboid is a solid that has six quadrilateral faces. When those quadrilaterals are rectangles, the shape is technically a rectangular prism. Each face meets another at right angles in the case of a true rectangular cuboid, but in the more general form (a parallelepiped), the faces can be parallelograms instead of rectangles.

Every cuboid has 6 faces, 12 edges, and 8 vertices. A cube is a special type of cuboid where all edges are equal, and each face is square. Rectangular boxes and bricks are everyday examples of cuboids.

Cuboids are especially important in geometry and real life because they help model the three-dimensional space we interact with daily. Volume and surface area calculations for cuboids are widely applied in packaging, architecture, and storage design.

Answer: l × w × h

The volume of a rectangular prism is calculated using the formula length × width × height. This measures the amount of three-dimensional space the prism occupies. Each dimension corresponds to one of the edges along the prism's length, width, and height.

Example:

A rectangular prism with a length of 5 units, width of 3 units, and height of 2 units has a volume of

5 × 3 × 2 = 30 cubic units.

This formula applies to all rectangular prisms, including cubes (where length = width = height).

Understanding this formula is important in many practical applications, such as packing boxes, calculating storage space, and designing containers. It's a foundational concept in geometry that also leads to volume formulas for other prisms and solids.