Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book.

To understand the dynamics of composite figures, one needs first to cover the basics of real two-dimensional geometric figures. Because at the end of the day, composite figures are the sum of basic figures.

Basic Properties of Two-Dimensional figures:

2 D Shapes	Properties
Triangle	It can have no, 2 or 3 equal sides	It can have no, 2 or 3 equal angles	It can have up to 2 axes of symmetry
Square	Four equal sides	Four equal angles(90°)	Four axes of symmetry
Rectangle	2 sets of 2 equal sides	Four equal angles(90°)	Two axes of symmetry
Circle	Constant diameter and radius	The total angle of a circle is equal to 360 degrees	Almost infinite axes of symmetry going through the center
Triangle	It can have no, 2 or 3 equal sides	It can have no, 2 or 3 equal angles	It can have up to 2 axes of symmetry
pentagon	5 sides (can be equal or unequal)	5 angles (can be equal or unequal)	It can have up to 5 axes of symmetry
hexagon	6 sides (can be equal or unequal)	6 angles (can be equal or unequal)	It can have up to 6 axes of symmetry
Octagon	8 sides (can be equal or unequal)	8 angles (can be equal or unequal)	It can have up to 8 axes of symmetry
Parallelogram	2 sets of 2 equal sides	2 sets of 2 equal angles	Usually no axes of symmetry
Rhombus	All sides are the same length	2 sets of 2 equal angles	2 lines of symmetry
Trapezium	At least 2 parallel sides	Can have pairs of equal angles	It can have a line of symmetry

Area of Composite Figures

The area of composite figures is the area that any composite shape covers. A composite shape comprises a few polygons that are joined together to form the desired shape. These shapes or figures can be made up of triangles, squares, and quadrilaterals, among other things. To calculate the area of a composite shape, divide it into basic shapes such as squares, triangles, rectangles, hexagons, etc.

A composite shape is made up of basic shapes that have been combined. It is also referred to as a “composite” or “complex” shape.

Now that we have figured out the basic properties of the important polygons, we can move into understanding the area and perimeter of the polygons:

2d Shape	Area	Perimeter
Circle	Πr2 (R is the radius of the circle)	2πr
Triangle	½ (Base x height)	Sum of three sides
Square	Side2	4(Side)
Rectangle	Length x Breadth	2(Length + Breadth)
Rhombus	½ (Product of diagonals)	4(Side)
Parallelogram	Base x Height	2 (Base + Side)

The volume of Composite Figures

Understanding volume when it comes to finding areas is crucial. You must know the volumes to calculate areas of various sections when limited data is provided. In mathematical terms, volume is the amount of space inside a 3-D object. First, we will understand some basic terms needed to find the volume of composite figures. Then we will get into the formulae.

Solid Figure: Three-dimensional figures of simple geometric shapes are called solid figures. Some examples of solid figures are cubes, prisms, pyramids, and cylinders. The volume of a Solid figure: The volume of a solid figure is the measure of the enclosed space within the figure.

Composite Figure: Figures constructed by connecting different solid figures are composite figures. The volume of a Composite Figure: Sum of the volumes of the given solid figures.

Let us understand the properties of some basic 3D objects like the way we discussed for the 2D objects:

3D Shapes	Properties
Sphere (With radius – r)	It has no edges or vertices (corners). It has one curved surface. It is perfectly symmetrical. All points on the surface of a sphere are at the same distance (r) from the center.
Cone	It has a flat base. It has one curved side and a one-pointed vertex at the top or bottom known as the apex.
Cylinder	It has a flat base and a flat top. The bases are always congruent and parallel. It has one curved side.
Cube	It has six faces in the shape of a square. The sides are of equal lengths.12 diagonals can be drawn on a cube.
Pyramid	A Pyramid is a polyhedron with a polygon base and an apex with straight lines. Based on their apex alignment with the center of the base, they can be classified into regular and oblique pyramids.
Prism	It has identical ends (polygonal) and flat faces. It has the same cross-section all along its length
Cuboid	It has six rectangular faces. All the sides of a cuboid are not equal in length.12 diagonals can be drawn on a cuboid.

How to Find Volume of a Composite Figure?

• Measure the dimensions of the bottom solid figure and find the volume.
• Measure the dimensions of the top solid figure and find the volume.
• Find the volume of the composite figure by adding the volume of the two solid figures.

Some of the important formulae regarding finding the surface areas and volumes of various composite figures:

3D Shape	Formulas
Cone	Curved Surface Area = πrl; (where ‘l’ is the slant height and l = √(h2 + r2))
Total Surface Area = πr(l + r)
Volume = (1/3)πr2h
Cylinder	Total Surface Area = 2πr(h+r); (where ‘r’ is the radius and ‘h’ is the height of the cylinder)
Volume = πr2h
Sphere	Diameter = 2 × r; (where ‘r’ is the radius)
Surface Area = 4πr2
Volume = (4/3)πr3
Cube	Lateral Surface Area = 4a2; (where ‘a’ is the side length of the cube)
Total Surface Area = 6a2
Volume = a3
Cuboid	Lateral Surface Area = 2h(l + w); (where ‘h’ is the height, ‘l’ is the length and ‘w’ is the width)
Total Surface Area = 2 (lw + wh + lh)
Volume = (l × w × h)
Pyramid	Surface Area = Base Area + (1/2 × Perimeter × Slant Height)
Volume = [(1/3) × Base Area × Altitude]
Prism	Surface Area = [(2 × Base Area) + (Perimeter × Height)]
Volume = (Base Area × Height)

How to find the area of a Composite Figure?

Finding the areas of composite figures is not much challenging. You have to memorize the formulae by heart and do the calculations. You can score more if you learn the formulae and practice them regularly. The area of the composite figures is the area of one or more simple polygons and circles combined. We can add the areas of all the basic figures together to calculate the area of the composite figures. Find the area of each shape and add them together to find the area of the composite figure. For example, the area of composite shapes is measured in m², cm², in², or ft².

Example:

1. Find the area of the composite figure when – 1. There is a rectangle of 4cm and 3cm in height & length, respectively. 2. A triangle with a 5cm base, 10cm height

      A: As we have learned in the past, a composite figure is a summation of basic polygons. So to find the area of composite figures, we have to break the figure into small polygons and then apply the formula as required.

Hence the area of the rectangular- 4 x 3 = 12 square cm

Area of the triangle – ½ x 10 x 5 = 25 square cm

Therefore, the total area of the composite figure = 12 + 25 = 37 square cm 

2. A composite figure has an area of 100 units square. The shape is composed of a circle and a triangle, and the area of the triangle is 64 units square. What is the area of the circle?

      A: Given the area of the composite figure = 100 units square and the area of the triangle = 64 units square

Using the formula for the area of the composite shape, Area of composite shape = area of triangle + area of the circle.

⇒ 100 = 64 + area of circle

⇒ Area of circle = 100 – 64

⇒ Area of the circle = 36 units square.

Therefore the area of the circle is 36 units square.

Frequently Asked Questions

1. What is a composite figure?

Ans. A composite figure is a drawing that represents multiple objects. Composite figures are often found in art, where they can be used to create the illusion of depth.

2. What is composite figure formula?

Ans. Composite figure formula is a formula to find the area of a composite figure. The formula is:

Area = (Area of rectangle) + (Area of triangle)

The rectangle is the area between the x-axis and y-axis, while the triangle is the area between the x-axis and line y=k.

3. What is a composite figure or shape?

Ans. A composite figure is a shape that is made up of two or more other shapes.It can be difficult to determine which shapes combine to make one composite shape, and this can be a problem for people with dyslexia.

4. How do you find the value of composite figures?

Ans. You can find the value of composite figures by adding together the values of their component parts. For example, if you have a composite figure made up of two triangles and one square, you add up the values of each triangle and then add them to the value of the square.

5. How do you find the length of a composite figure?

Ans. The length of a composite figure is the sum of all of its component parts. The formula for finding the length of a composite figure is length = ∑(length of each component)