Geometry – Math Tests – Practice Exercises & Quizzes

Arrows

Find the path

Basic 2D shapes and their locations

Basic 3D shapes

Lines of symmetry

Line Symmetry

What Is Line Symmetry?

Line symmetry is a property of shapes that look the same on both sides of an imaginary line – this line is called the line of symmetry.

Imagine that you fold a piece of paper exactly along the line of symmetry. If both halves of the shape match exactly, the shape has line symmetry.

Examples of Symmetrical Shapes

These shapes have lines of symmetry:

Square – has 4 lines of symmetry
Rectangle – has 2 lines of symmetry
Equilateral triangle – has 3 lines of symmetry
Circle – has infinitely many lines of symmetry

Symmetrical Letters

Some letters of the alphabet also have line symmetry:

A, M, T, U, V, W, Y – have a vertical line of symmetry
B, C, D, E, K – have a horizontal line of symmetry
H, I, O, X – have two lines of symmetry (vertical and horizontal)

Asymmetrical Shapes

Not every shape has a line of symmetry. For example:

An irregular polygon
The letters F, G, R, S, Z
Most natural objects (branches, stones…)

Summary

A line of symmetry is a line that divides a shape into two matching halves.
A shape can have one, several, or no lines of symmetry.
A circle has infinitely many lines of symmetry.
We can check symmetry by folding a piece of paper – both halves must match exactly.

Count the cubes (difficult)

Find the plan of a cube building

Plan View: Looking at a Structure from Above

A plan view is the technical term for a view from above. Imagine that you are a giant or flying in an airplane and looking straight down at a structure made of cubes.

1. What can we see in a plan view?

When looking from above, we cannot see how tall the columns are, but we can see exactly which squares of the grid are occupied and which are empty. The plan view shows us the structure's footprint.

2. Why is a plan view important?

From a single 3D picture, it may not always be clear what is behind a tall column. A plan view gives us an accurate picture of where the structure is located on the grid, including occupied positions that may be hidden in an ordinary 3D view.

Remember: A plan view is like a "map" of your structure!

2D shapes - sides and vertices

3D shapes - faces, vertices and edges

Perimeter of various shapes

Perimeter on a grid

Area on a grid

Compare perimeter and area of various shapes

Perimeter and area

Perimeter is the distance around the outside of a shape (green line).
Area is the amount of space (measured in square units, e.g. cm², m²) inside a shape (grey area).

Perimeter of rectangles and squares

Perimeter and area of rectangles and squares

Rectangle and squares are quadrilaterals with four right angles.

With squares, all four sides are the same length, so the perimeter is four times the length of a side a.

$$P = 4a$$

In a rectangle, two pairs of opposite sides are of equal length a, b. The perimeter of a rectangle is given by the formula:

$$P = 2a + 2b$$

The area is given by formulae:

Square:

$$A=a×a=a^2$$

Rectangle:

$$A = a×b $$

Area of rectangles and squares

Perimeter and area of rectangles and squares

Rectangle and squares are quadrilaterals with four right angles.

With squares, all four sides are the same length, so the perimeter is four times the length of a side a.

$$P = 4a$$

In a rectangle, two pairs of opposite sides are of equal length a, b. The perimeter of a rectangle is given by the formula:

$$P = 2a + 2b$$

The area is given by formulae:

Square:

$$A=a×a=a^2$$

Rectangle:

$$A = a×b $$

Triangle inequality

In each triangle, any two sides together are longer than the remaining third side. So, the following applies for their lengths in each type of triangle:

$$a+b>c$$

$$c+b>a$$

$$a+c>b$$

E.g. if we have the given lengths of two sides of the triangle a = 8, b = 3, c = 6, we can construct the triangle. But if the sides are a = 3, b = 4, c = 9, the triangle cannot be constructed because:

$$3 + 4 <9$$

If we know only two lengths (e.g. a = 3 and b = 7) and we have to find out the possible range of the length of the side c, we proceed as follows:

$$a + b> c ;→;3+7>c$$

$$;→; c <10$$

At the same time

$$ a+c > b ;→; 3+c > 7$$

$$ ;→; c > 4$$

And also:

$$b+c>a ;→;7+c>3$$

$$ ;→; c>0$$

So we know:

$$c<10$$

$$c>4$$

$$c>0$$

In order for c to satisfy all of these inequalities, its length must be between 4 and 10:

$$4

Area of a triangle on a grid (medium)

Area of a Triangle on a Square Grid

What is the area of the green and the blue triangle below?

If we look carefully, we notice that their area is exactly half of the rectangle into which they could fit:

The red rectangle has an area of 6 squares. The green and the blue triangle each have exactly half of that, so their area is 3 squares.

This rule works for every triangle. In the picture below, the green and the blue triangle also have an area equal to half of their rectangle — again 3 squares:

And even when the vertex of the triangle “sticks out” of the rectangle, its area is still half of it. That is why all the triangles in the picture below have the same area — half the area of the red rectangle:

So when we find the area of a triangle on a square grid, it does not have to be difficult at all. We just need to find the right rectangle or square that “belongs” to it. It is not a coincidence that the formula for the area of a triangle is: base times height to that base, all divided by two. By height we mean the perpendicular distance from the opposite vertex to the given side. The marked height of the purple triangle is 1 (so its area is 1 square), for the green triangle it is 2 (so its area is 2 squares), and for the blue triangle it is 3 (so its area is 1 and a half squares).

However, this method can only be used when one side of the triangle lies directly on the grid lines. But what about triangles that do not have such a side, like triangle ABC below?

If we subtract the areas of the three outer triangles ACD and CBF (each has an area of 1 and a half squares) and ABE (area 2 squares), we get the area of triangle ABC: 4 squares.

Altitudes (heights) of a triangle

Altitudes

An altitude (or the height) of a triangle is a line segment through a vertex and perpendicular to (forming a right angle with) a side of the triangle containing the base (the side opposite the vertex). This line containing the opposite side is called the extended base of the altitude. Every triangle has 3 altitudes which intersect at one point called the orthocenter (O).

In an acute triangle, all altitudes and the orthocenter lie within the triangle.

In a right triangle, two sides coincide with the altitudes and the orthocenter coincides with the vertex of the right angle.

In an obtuse triangle, two altitudes and the orthocenter lie outside of the triangle.

Volume and surface area of cubes and rectangular prisms

Cube and rectangular prism

A cube is a three-dimensional solid with 6 sides whose length, width, and height are equal. Its angles are right.

To calculate the volume and surface area, use these formulae:

$$V=a×a×a=a^3$$

$$S=2a×a+2a×a+2a×a=6a^2$$

A rectangular prism is a three-dimensional solid which has six faces that are rectangles. Its angles are right.

$$V=a×b×c$$

$$S=2a×b+2b×c+2a×c$$

Volume and surface area of cylinders, spheres and cones

Sphere

A sphere is a perfectly round solid figure with every point on its surface equidistant from its centre. The radius (r) of a sphere is the distance from the exact center of the sphere (C) to any point on the outside edge of that sphere. We calculate volume and surface area of a sphere using these formulae:

$$A=4πr^2$$

Surface area is actually equal to areas of four circles of the same radius.

Of all the solids, a sphere has the smallest surface area for a volume.

$$V = 4/3 πr^3$$

Cylinder

A cylinder is a solid that has two parallel (usually circular) bases connected by a curved surface.

The height (altitude, h) of a cylinder is the perpendicular distance between its bases.

A cylinder can be right or oblique. A right cylinder has bases aligned one directly above the other. In an oblique cylinder, the bases remain parallel to each other, but the sides lean over at an angle that is not 90°. If they have equal height and base area A_B, they will have the same volume.

$$V=A_B×height=πr^2h$$

The surface area is the sum of two bases A_B(pink) and the lateral area. We need to know the side length (s) to calculate the lateral area. Only for a right cylinder the side length is equal to the height (h=s₁).

For oblique cylinder it is s₂.

$$A=2×A_B+s×2πr=2πr^2+s×2πr$$

Cone

A cone is the solid figure that is generated by a right triangle turning around one of its cathetus. It can be right (the vertex is over the center of its base) or oblique (vertex is not over the center). The base can be a circle or an ellipse.

A cone is closely related to a pyramid and the formulae for their volume are similar (volume of a pyramid is one third of a prism with the same width, length and height and a cone is one third of a cylinder with the same base and height).

$$V=1/3×A_B×h$$

The slant height (s) of a right cone is the length of the segment from the vertex of the cone to the circle of the base. We don't define this for oblique cones.

The total surface area of a cone is the sum of the area of its base A_B and the lateral (side) surface A_L. The lateral surface of a right cone can be calculated as:

$$A_L=πr×s$$

Total surface area of right circular cone is:

$$A=A_B+A_L=πr^2+πrs=$$

$$=πr^2+πrs=πr(r+s)$$