Three Dimensional Shapes – JSS1 Mathematics Lesson Note (Third Term)

Solid figures are often called 3-dimensional shapes. A solid figure is simply anything that occupies space and has a definite shape.

Behavioural objectives

By the end of the lesson, students should be able to:

• define a face, an edge and a vertex of a solid;
• state the properties of common solid figures;
• explain what the surface area of a figure is;
• find the surface area of a triangular prism, a pyramid, a cylinder, a cube, a cuboid and a sphere.

What are faces, edges and vertices?

• Face: A face is the surface of a solid which is enclosed by edges.
• Edge: An edge of a solid is a line where two faces meet. An edge may be straight or curved.
• Vertex: A vertex of a solid is a point or corner where three or more edges meet.

What are the types of solid and their properties?

Do these

How do you find the surface area of a solid?

The surface area of a figure is the total area of all the sides of the figure.

Triangular prism (right-angled triangle)

To find the surface area of the prism above, follow the steps below.

1. Divide the figure into smaller shapes.
2. Find the area of each smaller shape.
3. Add the areas of the smaller shapes.

A right-angled triangular prism can be divided into five smaller shapes: two right-angled triangles and three rectangles.

Example

Find the surface area of the prism below.

Pyramid

A pyramid is an object that has a straight-sided base (for example a square, rectangular or triangular base) and triangular sides that meet at the top, called the apex.

Example

Cylinder

To find the surface area of a cylinder, follow the steps below.

1. Divide the figure into smaller shapes.
2. Find the area of each smaller shape.
3. Add the areas of the smaller shapes.

A cylinder can be divided into three parts: two circles and a curved surface.

The area of a circle is A = πr².

Since there are two circles in a cylinder, multiply the above formula by 2, that is:

The area of the circles is A = 2πr²

The curved surface area, C.S.A. = 2πrh

The surface area of a cylinder, A = Area of the circles + C.S.A.

A = 2πr² + 2πrh

A = 2πr(r + h)

Cube

A cube has 6 sides (faces) of the same surface area. Each side (face) of the cube is a square. Recall that the area of a square is A = L².

The surface area of a cube is A = 6L², where L is the length of a side of the cube (all the sides of a cube are the same length).

Cuboid

The surface area of a cuboid is A = 2Lw + 2Lh + 2wh, where:

• L is the length of the cuboid;
• w is the width of the cuboid;
• h is the height of the cuboid.

Example

Sphere

A sphere is a three-dimensional object (such as a ball or the earth) with every point on the surface at the same distance from the centre.

The surface area of a sphere is A = 4πr², where r is the radius of the sphere.

Evaluation

1. Define a face, an edge and a vertex.
2. Into which five smaller shapes can a right-angled triangular prism be divided?
3. Write the formula for the curved surface area of a cylinder and the formula for its total surface area.
4. Write the formula for the surface area of a cube and the formula for the surface area of a cuboid.
5. What is a sphere? Write the formula for its surface area.

Assignment

1. A cube has sides of length 4 cm. Find its surface area.
2. State what L, w and h stand for in the formula A = 2Lw + 2Lh + 2wh.