Perimeters of Compound Shapes
The perimeter of a shape is the total distance around its boundary. For compound shapes, which are made by joining simple shapes, find the perimeter by adding the lengths of all outer sides, ensuring not to include any internal boundaries shared between parts.
Start by identifying all the outer edges. If some lengths are missing, use given dimensions and properties (e.g., parallel sides, right angles) to find them.
Example: A shape formed by joining a rectangle 8 cm by 3 cm to a square 3 cm by 3 cm along one side. The perimeter is found by adding all outer sides: 8 + 3 + 3 + 3 + 8 = 25 cm.
- Add only the outer edges, not internal lines.
- Use given dimensions or calculate missing lengths using geometric properties.
- Check units and sum carefully.
Areas of Compound Shapes and Sections of Shapes
To find the area of a compound shape, divide it into simple shapes such as rectangles, triangles, and sectors of circles. Calculate each area separately, then add them.
For rectangles, area = length × width. For triangles, area = ½ × base × height. For circles, area = π × radius². For sectors (parts of circles), area = (θ/360) × π × r², where θ is the central angle in degrees.
Express areas involving circles or sectors in terms of π unless a decimal approximation is requested or specified by the question. This maintains exactness and is preferred in exams unless otherwise stated.
- Break compound shapes into simple shapes.
- Use correct formulas for each shape.
- Express answers in terms of π when circles or sectors are involved.
- Add all areas to find total area.
Surface Areas of Compound Solids
Surface area is the total area of all the outer surfaces of a solid. For compound solids, find the surface area by calculating the area of each face or curved surface and summing them.
For example, a solid made by joining a cylinder and a hemisphere requires calculating the curved surface area of the cylinder (excluding the base joined to the hemisphere) plus the curved surface area of the hemisphere.
Formulas to remember: Cylinder curved surface area = 2πrh, Sphere surface area = 4πr², Hemisphere surface area = 2πr² (curved part only). If the base of the hemisphere is exposed, include the base area πr² as well.
- Identify all exposed surfaces.
- Calculate areas of flat and curved surfaces separately.
- Sum all surface areas for total surface area.
- Express answers in terms of π when curved surfaces are involved.
Volumes of Compound Solids and Parts of Solids
Volume measures the space inside a solid. For compound solids, find the volume of each part and add them.
Common volume formulas: Cube: V = a³, Cylinder: V = πr²h, Sphere: V = 4/3 πr³, Hemisphere: V = ½ × (4/3 πr³) = 2/3 πr³.
When calculating the volume of a hemisphere, use the hemisphere formula directly. Express volumes involving π in terms of π unless decimal answers are requested or specified.
- Calculate volumes of individual parts using correct formulas.
- Add volumes for compound solids.
- Express volumes in terms of π when appropriate.
- Include units (e.g., cm³).
Worked Example: Area and Volume of a Compound Shape and Solid
Consider a shape made by joining a rectangle 10 cm by 6 cm and a semicircle of radius 3 cm attached along one 6 cm side.
To find the area: Calculate rectangle area = 10 × 6 = 60 cm². Calculate semicircle area = ½ × π × 3² = ½ × π × 9 = 4.5π cm². Total area = 60 + 4.5π cm².
If this shape is extruded to form a solid 5 cm deep, find the volume: Volume = area of cross-section × depth = (60 + 4.5π) × 5 = 300 + 22.5π cm³.
- Break down compound shapes into simple parts.
- Calculate each area or volume separately.
- Sum results and include units.
- Express answers in terms of π when circles or parts of circles are involved.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Perimeter of compound shape | Add lengths of all outer sides, exclude internal edges | Shape formed by rectangle and square: sum outer edges only | Check all sides, avoid double counting internal lines |
| Area of compound shape | Divide into simple shapes, calculate each area, sum | Rectangle plus semicircle: area = rectangle area + sector area | Express circle areas in terms of π unless decimal required |
| Surface area of compound solid | Calculate area of each exposed face or curved surface, sum | Cylinder with hemisphere on top: sum curved surface areas excluding joined base | Identify all exposed surfaces carefully |
| Volume of compound solid | Calculate volume of each part, sum | Cylinder plus hemisphere: volume = cylinder volume + hemisphere volume | Use correct formulas and express in terms of π when needed |
Active recall
Quick check
1. How do you find the perimeter of a compound shape made of a rectangle and a triangle joined along one side?
Add the lengths of all outer sides, excluding the side where the rectangle and triangle are joined.
2. What formula is used to find the area of a sector of a circle with central angle θ degrees and radius r?
Area = (θ/360) × π × r².
3. How do you calculate the volume of a hemisphere with radius r?
Volume = 2/3 × π × r³.
4. When calculating the surface area of a solid made by joining a cylinder and a hemisphere, what must you remember about the joined surfaces?
Do not include the area of the base of the hemisphere or the top base of the cylinder where they are joined, as these surfaces are not exposed.